Ansgar Belke
email: ansgar.belke@uni-due.de.

Ulrich Volz

Kenji Aramaki

George Chortareas

Volker Clausen

Laurent Ferrara

Andy Filardo

Maurizio Habib

Benedicto Lukanima

Toshiyuki Matsuura

Volker Nitsch

Tobias Stöhr

Franz Waldenberger

Martin Weale

Naoyuki Yoshino

Sébastien Lechevalier

Eiji Ogawa

Toshitaka Sekine

CEAFJP DISCUSSION PAPER #18-03 The Yen Exchange Rate and the Hollowing-out of the Japanese Industry

Keywords: Yen appreciation, exchange rates, Japanese manufacturing, hollowing out, hysteresis. JEL Classification: F31, O14

Since the demise of the Bretton Woods system, the Yen has seen several episodes of strong appreciation, including in the late 1970s, after the 1985 Plaza Agreement, the early and late 1990s and after 2008. These appreciations have not only been associated with "expensive Yen recessions" resulting from negative effects on exports; since the late 1980s, the strong Yen has also raised concerns about a de-industrialisation of the Japanese economy. Against this backdrop, the paper investigates the effects of changes to the Yen exchange rate on the hollowing out of the Japanese industrial sector. To this end, the paper uses both aggregate and industry-specific data to gauge the effects of Yen fluctuations on the output and exports of different Japanese industries, exploiting new data for industry-specific real effective exchange rates. Our findings support the view that the periods of yen appreciation had more than just transitory effects on Japanese manufacturing. The results also provide indication of hysteresis effects on manufacturing. While there are certainly also other factors that have contributed to a hollowing out of Japanese industry, a strong Yen played a role, too.

Introduction p.

2.

Literature review p.

3.

Empirical approach and findings p.

3.1. Analysis with annual data, 1970-2016 p.

3.1.1. Data and empirical model p.

3.1.2. Unit root tests p.

3.1.3. A cointegration approach: in search of long-run "causation" of the "hollowing out" p.

3.1.4. Autoregressive Distributed Lag (ARDL) estimations: in search of the dynamics underlying the "hollowing out" p.

3.2. Analysis with monthly data, 1991-2017 p.

3.2.1. Data and empirical model p.

3.2.2. Unit root tests p.

3.2.3. Sector-specific autoregressive distributed lag-estimations for Japanese industrial production p.

3.2.4. Panel estimations p.

Introduction

A strong Yen -endaka (円高) in Japanese -has repeatedly cause distress among Japanese policymakers and manufacturers. Since the demise of the Bretton Woods system in 1971, the Yen has seen several episodes of strong appreciation, including in the late 1970s, after the 1985 Plaza Agreement, the early and late 1990s and after 2008 (Figure 1). These appreciations have not only been associated with "expensive Yen recessions" -endaka fukyo (円高不況) in Japanese -resulting from negative effects on exports; since the late 1980s, the strong Yen has also raised concern about a de-industrialisation of the Japanese economy.

Indeed, the share of manufacturing in total output has declined from 26.7% in 1980 to 20.5% in 2015, while the share of manufacturing in total number of employed persons has declined from 27.0% to 16.2% between 1970 and 2016 (Figure 2). The decline in the share of manufacturing in total employment or GDP is not unique to Japan. It is indeed a phenomenon that can be seen across industrialised economies, with varying degrees. What makes Japan a particularly interesting case to analyse is that the Yen's real effective exchange rate has shown much larger volatility than the currencies of other large advanced economies. Between 1980 and 2017, the standard deviation of Japan's annual real effective exchange rate was 17.0, compared to 12.3 for the U.S., 10.4 for the United Kingdom, and 5.3 for Germany. Exchange rate effects on manufacturing may therefore have been more pronounced than elsewhere. While the strong Yen and its potentially de-industrialising effect have received much attention in the political and economic policy discourse in Japan, there has been surprisingly little research on this in the academic literature. As pointed out by Hamada and Okada (2009: 200), most research on Japan's "lost decade" and hollowing out of the Japanese industry "have been broadly focused on its real and domestic aspects, such as total factor productivity (TPF), growth decline, non-performing loans, and governance." Hamada and Okada (2009: 200) argue that monetary and exchange rate policy have played an important role in this, and that "Japanese industries endured a heavy burden" due to a greatly overvalued real exchange rate.

Against this backdrop, the paper aims to investigate the effects of Yen fluctuations on industrial production and the hollowing out of the Japanese industry. The major contribution of this study is an econometric analysis using new data for industry-specific real effective exchange rates (RIETI, 2018) to gauge the effects of Yen fluctuations on the output and exports of different Japanese industries at monthly frequency. As recently shown by [START_REF] Fort | New Perspectives on the Decline of US Manufacturing Employment[END_REF] for the US economy, changes to manufacturing employment have differed considerable across sectors.

The use of monthly, industry-specific data allows us to better capture potential effects of real effective exchange rate movements. We also use annual data since 1970 to investigate potential long-term effects of Yen exchange rate changes on manufacturing employment. Following Rowthorn and Coutts (2004: 767), we define de-industrialisation -to which we also refer as "hollowing out" -as "a secular decline in the share of manufacturing in national employment".

The topic is not only of relevance to the Japanese economy, it also bears relevance for other advanced economies. Indeed, recent years have seen an intense discussion in several member countries of the Eurozone about purported negative effects of a too high euro exchange rate on manufacturing and employment (e.g., [START_REF] Belke | The Strong Euro: Challenges for the European Central Bank and Implications for the Global Economy[END_REF]. For instance country outside the eurozone increases the value of the average exporter's sales to this country by around 5-6%", with most of this effect realised in the same year as the depreciation. Likewise, the U.S. has recently seen a renewed discussion on de-industrialisation, with a prominent role of the dollar exchange rate in this discourse (e.g., [START_REF] Campbell | Relative Prices, Hysteresis, and the Decline of American Manufacturing[END_REF][START_REF] Levinson | U.S. Manufacturing in International Perspective[END_REF]. A better understanding of any exchange rate effects on Japanese manufacturing should also provide insights for economic and monetary policy elsewhere.

The reminder of the paper is structured as follows. Section 2 provides a review of the literature on the effects of exchange rate changes on de-industrialisation. Section 3 comprises our empirical analysis.

Literature review

It is well established that depreciated real exchange rates can help to stimulate industrial development and economic growth [START_REF] Rodrik | The Real Exchange Rate and Economic Growth: Theory and Evidence[END_REF]. 1 Although there seems to exist a "fear of appreciation" [START_REF] Levy-Yeyati | Fear of Appreciation[END_REF] relatively little research has been conducted on the effect of overvalued exchange rates on economies. The first systematic empirical cross-country analysis to investigate the effects of large exchange rate appreciation on current account balances and on real output was conducted by [START_REF] Kappler | The Macroeconomic Effects of Large Exchange Rate Appreciations[END_REF], who find that episodes of strong exchange rate appreciations are associated with deteriorating current account balances and a slow-down of real export growth, but no significant effects on output. Likewise, [START_REF] Bussière | Currency Crises in Reverse: Do Large Real Exchange Rate Appreciations Matter for Growth?[END_REF] examine to what extent large and rapid real exchange rate appreciations impact on economic growth. Using a sample of 53 emerging and advanced economies, they find that while large appreciations dampen export growth and boost import growth, output growth is higher on average.

The literature on de-industrialisation has focused less on exchange rate valuations than on changes in specialisation, consumption, technological progress and productivity, international trade and investment patterns [START_REF] Rowthorn | De-industrialisation and the Balance of Payments in Advanced Economies[END_REF]. The major exemption is the U.S.

economy, for which the effect of the dollar exchange rate on the U.S. industry has been studied

widely. The findings in this literature are mixed, with some studies finding a large effect [START_REF] Branson | Dollar Appreciation and Manufacturing Employment and Output[END_REF], 1987, 1988a, 1988b;[START_REF] Revenga | Exporting Jobs? The Impact of Import Competition on Employment and Wages in U.S. Manufacturing[END_REF][START_REF] Gourinchas | Exchange Rates and Jobs: What Do We Learn from Job Flows?[END_REF]Kandil andMirzaie, 2002, 2003;[START_REF] Klein | Job Creation, Job Destruction, and the Real Exchange Rate[END_REF][START_REF] Campbell | Relative Prices, Hysteresis, and the Decline of American Manufacturing[END_REF] and others a small or no impact [START_REF] Glick | Does Exchange Rate Appreciation 'Deindustrialize' the Open Economy? A Critique of U.S. Evidence[END_REF][START_REF] Goldberg | Exchange Rates and Investment in United States Industry[END_REF]Campa and[START_REF] Campa | Investment in Manufacturing, Exchange Rates and External Exposure[END_REF][START_REF] Choi | Unit Root Tests for Panel Data[END_REF][START_REF] Bahmani-Oskooee | The Long-Run Effects of Depreciation of the Dollar on Sectoral Output[END_REF][START_REF] Bahmani-Oskooee | Sectoral Employment, Wages and the Exchange Rate: Evidence from the U.S[END_REF]. 2

1. A recent study by [START_REF] Alfaro | The Real Exchange Rate, Innovation and Productivity: Regional Heterogeneity, Asymmetries and Hysteresis[END_REF] confirms this for economies with high export participation.

2. There are also a number of studies investigating exchange rate effects on European economies. For instance, using firm-level data, [START_REF] Moser | International Competitiveness, Job Creation and Job Destruction -An Establishment-level Study of German Job Flows[END_REF] and Berman et al. (2012) find relatively small exchange rate effects on employment of German and French firms, respectively. [START_REF] Ekholm | Manufacturing Restructuring and the Role of Real Exchange Rate Shocks[END_REF] analyse the impact of a real appreciation of the Norwegian krona on Norwegian net-exporting and import-competing firms in the early 2000s, finding negative effects on domestic employment across both groups of firms.

CEAFJP DISCUSSION PAPER #18-03

According to Rowthorn andRamaswamy (1997a, 1997b) and Rowthorn andCoutts (2004, 2013), the most important internal/domestic factors contributing to deindustrialization are productivity growth in the manufacturing sector and shifting patterns of domestic expenditure, i.e., with rising income consumption tends to shifts towards more services and away from manufactured goods (known as Engel's Law or Bell's Law). The most important external factors are identified as "North-South" trade and the outsourcing of labour-intensive manufacturing to low-wage countries. However, domestic factors, especially a higher productivity growth in manufacturing than in services, are widely seen as the dominant factor behind deindustrialization (Rowthorn and Ramaswamy, 1997; Rowthorn and Coutts, 2004, 2013; Lawrence, 2017). 3 Surprisingly few studies have looked into the potentially de-industrialising effect of endaka, even though [START_REF] Obstfeld | Time of Troubles: The Yen and Japan's Economy[END_REF] pointed out that Japan's real economic growth rate has been strongly negatively correlated with the level of the Yen's real effective exchange rate. Only few studies can be found that systematically try to verify the hypothesis that the strong Yen has contributed to de-industrialisation and outsourcing of industry in Japan, including [START_REF] Dekle | Endaka and Japanese Employment Adjustment[END_REF], [START_REF] Dekle | The Japan-US Exchange Rate, Productivity, and the Competitiveness of Japanese Industries[END_REF] and [START_REF] Yamashita | Exchange Rates and Deindustrialization: Japanese Experiences[END_REF][START_REF] Yamashita | Does 'Driving Range' Really Matter? The Hidden Cost of Internal Combustion Engine Vehicles and the Potentially New Value Proposition of Electric Vehicles: Two Cases from Countryside and Urban Side of Japan[END_REF]. [START_REF] Dekle | Endaka and Japanese Employment Adjustment[END_REF] and [START_REF] Dekle | The Japan-US Exchange Rate, Productivity, and the Competitiveness of Japanese Industries[END_REF] find that Yen appreciations of 1985 and 1995 significantly hurt the ability of Japan to compete with the U.S. by raising the relative production costs of Japanese industries. [START_REF] Yamashita | Exchange Rates and Deindustrialization: Japanese Experiences[END_REF][START_REF] Yamashita | Does 'Driving Range' Really Matter? The Hidden Cost of Internal Combustion Engine Vehicles and the Potentially New Value Proposition of Electric Vehicles: Two Cases from Countryside and Urban Side of Japan[END_REF] finds that Yen appreciation and outward FDI contributed to deindustrialization and suggests that a weak Yen may contribute to reshoring. 4

A string of studies examines the role of Japanese outward direct investment in the hollowing out process. Japanese outward direct investment took off in the mid-1980s [START_REF] Fukao | Vertical Intra-industry Trade and Foreign Direct Investment in East Asia[END_REF]. The number of Japanese multinationals increased by 290% between 1985 and 1992, while overseas production of Japanese firms increased from 3% in 1982 to 17% in 2002 [START_REF] Ryan | Hollowing Out of the Japanese Economy: A Long-term Perspective[END_REF]. Analysing data on Japanese acquisitions in the United States from 1975-1992[START_REF] Bloningen | Firm-Specific Assets and the Link between Exchange Rates and Foreign Direct Investment[END_REF] finds that real dollar depreciations increased the likelihood of Japanese acquisitions in U.S. industries. Likewise, the BOJ (2011) attributed the renewed shifting of production abroad since 2008 to the relative appreciation of the Yen compared to the levels before the Lehman shock.

Empirically, the effect of outsourcing on domestic manufacturing employment is mixed.

According to [START_REF] Baldwin | Globalisation: The Great Unbundling(s)[END_REF], offshoring labour-intensive production processes to adjacent lower-cost locations in Southeast and Northeast Asia since the mid-1980s did apparently allow Japanese companies to retain parts of manufacturing production domestically while maintaining international competitiveness. The BOJ (1989) finds that increased outward direct investment contributed to long-term structural adjustment in the export sector. This contributed to a temporary surge in the export of capital goods and parts to overseas production bases of Japanese companies. [START_REF] Yamashita | Expansion Abroad and Jobs at Home: Evidence from Japanese Multinational Enterprises[END_REF] find no indication that expansion of overseas operations by Japanese manufacturing multinational enterprises reduced home employment [START_REF] Thorbecke | Exchange rates and the Swiss economy[END_REF] examine the effect of changes to the Swiss franc exchange rate on the Swiss economy and find differences related to the sophistication of exports. 3. A further factor that appears to have played a role in the fallowing share of manufacturing is the outsourcing of various services that historically were conducted in-house and hence counted as manufacturing, whereas they are now counted as services. See [START_REF] Boulhol | Deindustrialisation and the Fear of Relocations in the Industry[END_REF]. Evidence for this development in Japan is presented in [START_REF] Uemura | The Transformation of Growth Regime and Deindustrialization in Japan[END_REF]. 4. [START_REF] Bleaney | Invoicing-currency Effects in the Pricing of Japanese Exports of Manufactures[END_REF] looks at the pricing of Japanese manufacturing exports and finds a significant role of the yen real exchange rate.

CEAFJP DISCUSSION PAPER #18-03

over the period 1991-2002; indeed, their analysis suggests that it may have contributed to maintaining the level of home employment. A recent study by [START_REF] Ryan | Hollowing Out of the Japanese Economy: A Long-term Perspective[END_REF] uses Japanese firm-level data for the period 1982-2001 to compare employment between firms with and without foreign affiliates and the extent to which the development of foreign operations affects firm-level domestic employment. It finds "some limited evidence" of the hollowing out of the Japanese economy by Japanese multinational firms shifting production abroad from 1991 onwards.

Using the same industry-specific exchange rates that will be employed in our empirical analysis later, [START_REF] Sato | Industryspecific Real Effective Exchange Rates and Export Price Competitiveness: The Cases of Japan, China, and Korea[END_REF] analyse export price competitiveness and export performance of Japan, China and Korea and find that the strong Yen after 2007 destroyed the Japanese electronics sector. Likewise, [START_REF] Thorbecke | The Short-and Long-Run Effects of Exchange Rate Changes on the Japanese Electronics Industry[END_REF] finds in a study focused on the Japanese electronics industry that the Yen appreciation between 2007 and 2011 had a very damaging effect on electronics exports. However, neither study controls for employment effects. [START_REF] Kato | Exports, Exchange Rates, and Productivity: An analysis of the Japanese Manufacturing Sectors[END_REF] examines the effects of exchange rate changes and productivity on manufacturing exports for the period 2002-2012 and finds exchange rates to be important factors to affect firmlevel exports.

An important issue in this context is that exchange rate appreciations and depreciations may not have symmetrical effects. That is, a period of strong appreciation may lead to outsourcing and the loss of manufacturing capacities -and employment -in the home country, but a depreciation later on may not bring about a return (or "re-shoring") of manufacturing activities and employment, given that long-term investments have been made abroad, which

will not be reversed. These hysteresis effects have been described in sunk cost models of trade hysteresis [START_REF] Krugman | Deindustrialization, Reindustrialization, and the Real Exchange Rate[END_REF][START_REF] Baldwin | Persistent Trade Effects of Large Exchange Rate Shocks[END_REF][START_REF] Baldwin | Hysteresis in Trade[END_REF][START_REF] Baldwin | Exchange Rate Hysteresis: Large versus Small Policy Misalignments[END_REF] and empirically investigated mostly for the U.S. and a number of European economies. 5 Giovannetti and Samiei (1995) conduct an empirical analysis of the importance of hysteresis in international trade, using manufacturing exports for the United States, Germany, and Japan.

They find "strong evidence" in favour of the presence of hysteresis only in the case of Japanese exports.

Empirical approach and findings

For our econometric analysis, we use both aggregate and industry-specific data to gauge the effects of changes of the real effective Yen exchange rate. The selection of explanatory variables in ( 1) is motivated by the standard variables used in previous studies on de-industrialisation. To these we add EXR as the variable of our particular interest to investigate the hypothesis that movements of the Yen's real effective exchange rate have contributed to a de-industrialisation of the Japanese economy.

Increases in fixed capital formation could theoretically have both positive and negative effects on manufacturing employment. On the one hand, an increased investment in the domestic manufacturing sector could have positive employment effects. On the other hand, however, capital investment could advance the automation of manufacturing production, with neutral or even negative effects on manufacturing employment. In contrast, an increase in manufacturing exports should clearly have a positive effect on domestic manufacturing employment. As discussed above, the effect of outward direct investment can be ambiguous: on the one hand a shifting of production abroad may have negative employment effects, but at the same time there could be positive outcomes because it may increase the competitiveness of domestic operations which would remain important parts of regional or global value chains [START_REF] Baldwin | Globalisation: The Great Unbundling(s)[END_REF][START_REF] Yamashita | Expansion Abroad and Jobs at Home: Evidence from Japanese Multinational Enterprises[END_REF].

As mentioned above, productivity increases have been identified in the de-industrialisation literature as a major factor behind the fall in manufacturing employment, so we would expect a negative coefficient. We use TFP as our measure of productivity. We considered using alternative measures of productivity, such as multifactor productivity, but were not able to obtain data ranging back to the 1970s.

The effects of Chinese export growth are, again, ambiguous. 7 While Chinese export growth reflects of China's rapid growth process, which has also opened up great opportunities for Japanese companies, Chinese exporters have increasingly become competitors of Japanese manufacturing firms. Last but not least, we would expect the coefficient estimate for the real effective exchange rate -our main variable of interest -to be negative if the hollowing out hypothesis were to hold.

6. The same framework is applied by [START_REF] Boulhol | Deindustrialisation and the Fear of Relocations in the Industry[END_REF], who analyse the change in the share of manufacturing in employment between 1970 and 2002 for 16 OECD countries. But none of these studies controls for exchange rate effects. 7. Unfortunately, no data was available for Chinese manufacturing exports for this period. This is why we selected total exports from China.

CEAFJP DISCUSSION PAPER #18-03 At first glance, all timeseries appear to contain, probably stochastic, trends. We will explicitly check for stochastic trends before we start our estimation exercise in this section in order to make sure that the stochastic properties of the included variables meet the standard assumptions of our regression analysis. Moreover, some first indications of correlations can be derived as well. Whether this means "causation" in an econometric sense will be checked later on in this section.

The top left panel of Figure 3 shows the two main variables of interest, the share of manufacturing in total number of employed persons and the real effective exchange rate.

Between 1970 and the mid-1990s, the real effective exchange rate showed a clear long-term appreciation trend, but with two periods -1978-1982 and 1988-1990 -of

Unit root tests

We proceed with unit root tests of the variables to be employed in our empirical model.

The results of our single time series unit root tests are displayed in Table 1. According to Table 1, all variables turn out to be integrated of order one (I(1)), except Chinese exports EXPCHIN which appears borderline I(2). As a robustness check, we also conducted Dickey-Fuller GLS tests, Phillips-Perron tests, Kwiatkowski-Phillips-Schmidt-Shin (KPSS) and Ng-Perron unit root tests not only for the Yen exchange rate but also for all other variables. The results overall stayed the same. Among others, they support treating EXPCHIN finally as an I(1) variable in our regressions.

Given the unit root tests conducted in this section and their rather unambiguous results, we feel legitimised to employ the levels of the variables in the framework of a cointegration approach (option 1) and/or use the first differences of all variables as stationary variables in our Autoregressive Distributed Lag (ARDL) estimations (option 2).

Regarding our regression analysis two issues come to mind. Firstly, there is a relatively broad consensus in the literature that one may well find impacts of exchange rate changes on employment even if there is less evidence of an impact on trade [START_REF] Belke | Real Impacts of Intra-European Exchange Rate Variability: A Case for EMU?[END_REF].

Secondly, a thorough regression analysis is necessary to consider leads and lags of the variables when it comes to establish any one-way "causation". This is dramatically shown by the main results of the export hysteresis literature which shows that a positive coefficient of the exchange rate in export, employment, investment or industrial output equations may just hide a more complex dynamic of the relationship which may include stepwise negative effects of an appreciation of the home currency [START_REF] Krugman | The Persistence of the US Trade Deficit[END_REF][START_REF] Belke | Exchange Rate Bands of Inaction and Play-Hysteresis in German Exports -Sectoral Evidence for Some OECD Destinations[END_REF]). 8

8. The hysteresis argument is mainly related to the irreversibility of investment-type decisions, due to high fixed costs which cannot be recovered ex post (sunk costs). Industrial manufacturing production, the

CEAFJP DISCUSSION PAPER #18-03

We now turn to the first option, the estimation of a cointegrating equation in order to assess the impact of the Yen exchange rate (EXR) on the share of manufacturing in total number of employed persons in Japan (EMPMAN), taking the other variables contained in eq. ( 1) into account simultaneously in the long-run relationship.

A cointegration approach: in search of long-run "causation" of the "hollowing out"

As our cointegration framework we choose Dynamic Ordinary Least Squares (DOLS).

This is because DOLS allows us to test for causality running from the independent to the dependent variables. The DOLS estimation technique controls for endogeneity of explanatory variables [START_REF] Stock | A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems[END_REF][START_REF] Wooldridge | Introductory Econometrics: A Modern Approach, 4 th Edition[END_REF]. Endogeneity in the form of feedback effects or reverse causality between the dependent and independent variables would lead to a misspecification of our estimated models. More specifically, the DOLS procedure controls

for endogeneity of all explanatory variables by inserting leads and lags of the changes of all exogenous variables. Hence, DOLS is a powerful estimation technique, where standard errors are corrected for heteroscedasticity and cross-section correlation (Saikkonen, 1991;[START_REF] Stock | A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems[END_REF]. By inserting the leads and lags of the exogenous variables in first differences, the explanatory variables in levels become (super-) exogenous and the regression results

unbiased [START_REF] Wooldridge | Introductory Econometrics: A Modern Approach, 4 th Edition[END_REF]. Table 2 displays the results of a DOLS estimation of our fully specified empirical model (eq.

(1)). The share of manufacturing in total number of employed persons (EMPMAN) represents the dependent variable. The estimated coefficient of the Yen exchange rate (marked in bold letters in Tab. 3) is negative as expected from theory and highly significant. Seen on the whole, also the other estimated coefficients are significant and their signs corresponding with our priors. TFP shows a negative coefficient, suggesting that higher productivity has resulted in a focus of the second part of this paper, is a very good example because it is dependent on investmenttype decisions related to, among others, fixed investment, hiring and firing decisions and establishing distribution networks.

CEAFJP DISCUSSION PAPER #18-03

lower share of manufacturing employment, but this effect is not statistically significant. Gross domestic fixed capital formation shows a positive sign and is highly significant. As expected, manufacturing exports have a positive and significant effect on manufacturing employment, whereas outward direct investment has a negative and significant effect. Chinese exports show a positive sign, suggesting that the positive effects of Chinese economic development on Japanese manufacturing dominated the negative competition effect. Finally, the empirical realisations of the goodness-of-fit criteria, among them the very high R-Squared, indicate the appropriateness of our selected empirical model.

Our robustness checks reveal that the total goodness-of-fit does not become significantly lower if EXPCHIN is eliminated from our empirical model. This does not come as a surprise since both variables represent indicators of the world business cycle, i.e., the so-called "global factor". For exactly this reason, we leave out EXPCHIN in our next DOLS-specification. The corresponding regression results are displayed in Table 3. Again, the estimation results clearly are in accordance with our priors. As robustness checks we also conducted DOLS regressions with White correction of the coefficient covariance matrix [START_REF] White | A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity[END_REF]. There is no substantial change of the estimation results, with respect to the magnitude and the sign of the estimated coefficients. If at all, the significance of, for instance, the Yen exchange rate, even slightly increases once again in the specification excluding Chinese exports. Moreover, we also enacted DOLS regression with Newey-West correction of the coefficient covariance matrix [START_REF] Newey | A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix[END_REF]. Again, the estimation results do not change much with respect to both the magnitude and the sign of the estimated coefficients. All these results are available on request.

Application of the DOLS procedure requires the time series to be non-stationary and to be cointegrated over time. We are able to show that both conditions are satisfied in our case.

Evidence of non-stationarity has been delivered already by the results of our unit root tests in section 3.1.2. And empirical support of cointegration among the variables contained in our empirical models is conveyed by cointegration tests (Table 4 for the fully specified model and Table 5 for the model without Chinese exports). Figures 4 and5 display the residuals (which form the basis for the Engle-Granger cointegration tests described in Tables 4 and5) and the actual and fitted values according to the DOLS regression models described in Tables 2 and3, respectively. We now turn to the second option mentioned above, that is to conduct ARDL estimations based on the first differences of the non-stationary variables.

CEAFJP DISCUSSION PAPER #18-03 3.1.4. Autoregressive Distributed Lag (ARDL) estimations: in search of the dynamics underlying the "hollowing out"

In this section, we employ ARDL estimations in order to estimate the intertemporal dynamics of the equation explaining the change in manufacturing employment (D(EMPMAN)).

ARDLs are standard least squares regressions that include lags of both the dependent variable and explanatory variables as regressors [START_REF] Greene | Econometric Analysis, 6 th Edition[END_REF]. Their application requires the data to be stationary. This leads us to employ (only) the first differences of the variables of our empirical model. This comes at the cost that we are not able to exploit level information (as it is the case in our cointegration exercise). The ARDL model selection process employed by us uses the same sample for each estimation and selects the final model by maximising the empirical realisation of information criteria (in our case, of the Akaike criterion). The empirical results of the ARDL exercise is displayed in Table 6. The expression ARDL(1, 2, 0, 1, 1, 0, 0) in Table 6 lags of the Yen exchange rate (EXR). The result column containing the Yen exchange rate effect denotes the significant estimated coefficient for the exchange rate (-0.016393) and its significance level (0.02) in the selected ARDL model. Values in brackets in Table 6 denote the respective significance level (probabilities). Once again, we find a positive and significant effect of the Yen exchange rate on the share of manufacturing employment.

Summing up our annual data analysis, the DOLS and ARDL estimations provide robust results which indicate that increases in the real effective Yen exchange rate, i.e., appreciations, did have significant negative effects on the share of manufacturing in total employment in Japan. This is despite the fact that the Yen also experiences longer periods of real effective depreciation and is thus indicative of hysteresis effects [START_REF] Krugman | Deindustrialization, Reindustrialization, and the Real Exchange Rate[END_REF]Baldwin, 1987, Belke et al., 2013). Our DOLS estimations also find a negative and significant effect of outward FDI.

Analysis with monthly data, 1991-2017

Having investigated the long-term impact of real effective exchange movements on aggregate manufacturing employment with annual data spanning almost five decades, we now turn to an analysis using higher frequency and, importantly, sector specific data. If exchange rate changes are to have a long-term impact on manufacturing, there ought to be some shortterm impacts too. Using sector-specific data at higher frequency allows us to uncover such potential effects. Ideally, we would be using sector-specific employment to mirror the annual data analysis. However, since we were not able to obtain matching monthly sector-specific employment data, we chose to use sector-specific industrial production data as dependent variable as a second-best option, given industrial production and industrial employment are highly correlated.

Data and empirical model

In order to exploit our new dataset on sector-specific Yen exchange rates (Figure 6), which are calculated by using industry-specific producer price indices (RIETI, 2018), we now estimate the real Yen effective exchange rate effects on Japanese industrial production, using monthly data from 1991 to 2017:

(

Unit root tests

We now proceed with unit root tests of the variables listed below eq. ( 2) to be employed in our ARDL estimations and in our empirical mixed time series-panel models of Japanese industrial production which both necessitate the variables to be stationary.

The wide array of unit root tests we conducted, such as the ordinary Augmented-Dickey- As a first step, we test for unit roots in the pooled sector-specific time series. In order to make sure that we employ stationary variables in our panel estimations we conducted the following panel unit root tests for the pools of sector-specific variables: [START_REF] Levin | Unit Root Tests in Panel Data: Asymptotic and Finite-Sample Properties[END_REF], [START_REF] Breitung | The Local Power of Some Unit Root Tests for Panel Data[END_REF], [START_REF] Im | Testing for Unit Roots in Heterogeneous Panels[END_REF], Fisher-type tests using ADF and PP tests [START_REF] Maddala | A Comparative Study of Unit Root Tests with Panel Data and a New Simple Test[END_REF][START_REF] Choi | Unit Root Tests for Panel Data[END_REF][START_REF] Hadri | Testing for Stationarity in Heterogeneous Panel Data[END_REF]. All variables turned out to be I(1). The results are available on request.

As a second step, we conduct Pooled Least Squares estimations of a mixed time seriescross section model based on stationary time series with White cross-section standard errors (to allow for general contemporaneous correlation between the branch-specific residuals) and

White covariance (White, 1980, MacKinnon and[START_REF] Mackinnon | Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties[END_REF]. Non-zero covariances are allowed across cross-sections (degree-of-freedom corrected). The estimator we employ in this study is thus robust to cross-equation (contemporaneous) correlation between the branch-specific residuals and heteroskedasticity. The estimation results for our final model are displayed in Table 11.

CEAFJP DISCUSSION PAPER #18-03

As a third step, we conduct Pooled Estimated Generalised Least Squares (EGLS, crosssection weights), again with White cross-section standard errors and White covariance. The results are summarised in Table 12. The results of our fixed effects redundancy test (at the beginning of Table 12) support our specific choice of fixed effects empirically. And as a fourth and final step, we present estimation results based on Pooled Estimated Generalised Least Squares (EGLS) regressions conducted by the cross-section Seemingly Unrelated Regression (SUR) method (Table 13). 10 Also in this case, we rely on White crosssection standard errors and White covariance. Again, the tabulated results of the fixed effects redundancy test (in the upper part of Table 13) empirically corroborate our specific choice of 10. Cross-section SUR allows for contemporaneous correlation between cross-sections (clustering by period). It is sometimes referred to as the Parks estimator.

CEAFJP DISCUSSION PAPER #18-03 fixed effects. In all three estimation variants applied here we allow not only for fixed effects in the constant but also for cross-section specific slope coefficients. The selection of the final model was conducted according to the same criteria applied throughout the paper and described in detail beforehand. Overall, the estimation results displayed in Tables 11 to 13 again appear well behaved with an eye on our main hypothesis. Significant negative sector-specific impacts of the Yen effective real exchange rate can be established for at least six and a maximum of all nine sectors, the exact number of sectors depending on the chosen estimation procedure. This corroborates the empirical findings of [START_REF] Sato | Industryspecific Real Effective Exchange Rates and Export Price Competitiveness: The Cases of Japan, China, and Korea[END_REF] and [START_REF] Thorbecke | The Short-and Long-Run Effects of Exchange Rate Changes on the Japanese Electronics Industry[END_REF] for Japanese exports.

According to our Pooled Least Squares estimations (Table 11) significantly negative effects of the (sector-specific) Yen exchange rate on Japanese industrial production emerge for the chemicals, the optical instruments, the paper, the rubber, the wood and the textiles sectors.

CEAFJP DISCUSSION PAPER #18-03

Conclusions

In this paper we have investigated the role of the Yen exchange rate in the de-industrialisation that has taken place in Japan over recent decades. There are certainly multiple factors that have played a role in this process, and we do not seek to evoke the impression that the exchange rate has been the single most important factor. Yet our results suggest that a strong Yen had more than just transitory effects on Japanese manufacturing employment and output.

In our annual data analysis, the DOLS and ARDL estimations provide robust results which indicate that appreciations of the real effective Yen exchange rate did have significant negative effects on the share of manufacturing in total employment in Japan. This is despite the fact that the Yen also experienced longer periods of real effective depreciation, which is indicative of hysteresis effects on manufacturing. Our findings are consistent with recent research on the hollowing out of the U.S. economy, where findings by Campbell ( 2017) also point at the presence of hysteresis effects. But we should also point out that the magnitude of the exchange rate effects in our annual data analysis are much smaller than of those variables that have the biggest effect on manufacturing employment, namely TFP and fixed capital formation.

We also find significant negative effects of the real effective Yen exchange rate on industrial output when using monthly data. Our ARDL estimations find significant negative effects for chemicals, electrical equipment, transport equipment, rubber, optical instruments and paper.

Our panel analysis with sector-specific monthly data suggests that movements of the sectorspecific real effective Yen exchange rate had significant impact on up to seven industrial sectors (chemicals, optical instruments, rubber, wood, textiles, paper and transport equipment). Still, we should also mention that our ARDL and panel estimations did not uncover significant exchange rate effects for several industrial sectors, including general machinery, one of the most important industrial sectors for Japan.

From a policy perspective, our findings suggest that central banks should try to avoid rapid, large-scale appreciations of the type Japan has seen a number of times over recent decades as negative effects cannot be easily reversed at a later stage. Yet Japan may also be special in the sense that its geographical proximity to emerging economies across East Asia facilitated outsourcing.

Figure 1 :Figure 2 :

 12 Figure 1: Real effective exchange rate of the Japanese Yen, 1964-2017 (2010=100)

  with EMPMAN = Share of manufacturing in the total number of employed persons (%), CAP = Gross domestic fixed capital formation (% of GDP), EXPMAN = Manufacturing exports, OFDI = Outward direct investment, TFP = Total factor productivity at constant national prices, EXPCHIN = Chinese exports, and EXR = Real effective exchange rate. Appendix 1 provides more details on the variables and sources.

Figure 3 :

 3 Figure 3: Variables for analysis with annual data

Figure 3

 3 Figure 3 graphically displays the time series of the variables used in our estimations based on annual data. It conveys a first impression of the stationarity properties of the time series used. At first glance, all timeseries appear to contain, probably stochastic, trends. We

  depreciation. After a peak in 1995, the real effective exchange rate then shows a declining trend, albeit with periods of strong appreciation in 1998-2000, 2007-2011 and 2015-2016. Over the entire sample period, CEAFJP DISCUSSION PAPER #18-03the share of manufacturing in total number of employed persons shows a long-term declining trend, although the share of manufacturing remains stable over the periods1987-1992, 2005- 2007 and 2012-2016.

For

  the real effective Yen exchange rate EXR, all complementary unit root tests such as the Dickey-Fuller GLS (ERS) test, the Phillips-Perron test, the Kwiatowski-Phillips-Schmidt-Shin test and the Ng-Perron unit root test) except the ADF-test point into the direction of an I(1) process. Hence, we proceed with the Yen exchange rate considering it as an I(1) variable. The results are available on request.

Figure 4 :Figure 5 :

 45 Figure 4: Residuals, actual and fitted values (fully specified DOLS model)

  indicates the model selected by our search algorithm according to the Akaike information criterion. The first entry in brackets denotes the lags of the endogenous lagged variable. The second entry corresponds with the lag number of the change of total factor productivity, the third entry indicates the lag number of the change in gross domestic fixed capital formation (CAP). The fourth one stands for the lag number of the change in manufacturing exports (EXPMAN). The fifth entry denotes the lag number of the change in Chinese exports (EXPCHIN). The sixth entry is related to the lag number of the outward direct investment variable (OFDI). And the seventh entry refers to the lag number of CEAFJP DISCUSSION PAPER #18-03

  2) INP_X = β 0 + β 1 INP_X t-1 + β 2 EX_X + β 3 REER_X + β 4 INDINPUTPRICE + β 5 TFP + β 6 INPUSA + ε, where INP_X = Industrial production by industry X, EX_X = Exports by industry X, REER_X = Industry-specific real effective exchange rate of industry X for 11 different industrial sectors and manufacturing as a whole, INDINPUTPRICE = Industrial input price, and INPUSA = Industrial production United States. 9 Appendix 2 provides further details on the variables and sources used.Our data set covers the following 11 industrial sectors: chemicals (CHEMICALS), electrical equipment (ELECTRICALEQUIPMENT), food (FOOD), general machinery (GENERALMACHINERY), optical instruments (OPTICALINSTRUMENTS), paper (PAPER), petroleum related products (PETROLEUM), rubber related products (RUBBER), textiles (TEXTILES), transport equipment (TRANSPORTEQUIPMENT), wood related products (WOOD) 9. The time series of industrial production have been seasonally adjusted, using the Census X-13 procedure.CEAFJP DISCUSSION PAPER #18-03and manufacturing as a whole (MANUFACTURINGALL). Details of these sectors are provided in Appendix 3. While all other variables are industry-specific, INDINPUTPRICE and INPUSA are nonindustry-specific variables. In this sense, we follow a mixed panel-time series modelling approach after having estimated sector-specific ARDL models. Figure7shows industrial production (INP) and industry-specific real effective exchange rates (REER) for selected industries. The two series tend to move in opposite direction, indicating that real effective exchange rates may indeed have a negative impact on industrial production.

Figure 6 :Figure 7 :

 67 Figure 6: Real effective exchange rates for different industrial sectors

  Fuller test, the Dickey-Fuller GLS (ERS) test, the Phillips-Perron test, the Kwiatowski-Phillips-Schmidt-Shin test and the Ng-Perron unit root test, point into the direction of an I(1

  The empirical part is split into two sections: Firstly, we conduct an empirical analysis with annual, aggregate data from 1970 to 2016 that allows us to investigate potential long-term effects of Yen exchange rate changes on manufacturing employment. Secondly, we use monthly, industry-specific data from January 2001 to June 2017 to gauge the dynamics of Yen exchange rate changes on Japanese sectoral manufacturing employment. Section 4 concludes.

18-03 3.1. Analysis with annual data, 1970-2016 3.1.1. Data and empirical model

  Building onRowthorn and Ramaswamy (1997a) andRowthorn and Coutts (2004, 2013), we estimate the following model for Japan and annual data ranging from 1970 to 2016: 6

	(1) EMPMAN = β 0 + β 1 EMPMAN t-1 + β 2 CAP + β 3 EXPMAN + β 4 OFDI + β 5 TFP + β 6 EXPCHIN
	+ β 7 EXR + ε,
	Taking account of the availability
	of different data series, we firstly conduct an empirical analysis with annual data from 1970 to
	2016 that allows us to investigate potential long-term effects of Yen exchange rate changes
	on manufacturing employment. Secondly, we use monthly data from 2001:M01 to 2017:M06
	to check the dynamics of Yen exchange rate changes on Japanese sectoral manufacturing
	employment, exploiting a new monthly dataset on Japanese sectoral exchange rates which

are only available from 2001 onwards. For this purpose, we use a mixed time series-panel framework.

5. See

[START_REF] Belke | Exchange Rate Bands of Inaction and Play-Hysteresis in German Exports -Sectoral Evidence for Some OECD Destinations[END_REF]

,

Belke and Kronen (2018)

,

[START_REF] Campa | Exchange Rates and Trade: How Important is Hysteresis in Trade?[END_REF]

and

[START_REF] Campbell | Relative Prices, Hysteresis, and the Decline of American Manufacturing[END_REF]

.

CEAFJP DISCUSSION PAPER #

Table 1 : Results of unit root tests (ADF test statistics and probabilities)

 1

Source: Compiled by authors. Note: Table displays empirical realisations of the Augmented Dickey-Fuller test with a constant (MacKinnon (1996) one-sided p-values).

Table 2 : Dynamic least squares estimation for fully specified model

 2

Source: Compiled by authors.

Table 3 : Dynamic least squares estimation for model excluding Chinese exports Source

 3

: Compiled by authors.

Table 6 : ARDL estimation of fully specified model

 6

Source: Compiled by authors.

Table 11 : Sectoral industrial production in Japan -determinants according to a Pooled Least Squares panel model

 11

Source: Compiled by authors.

Table 12 : Sectoral industrial production in Japan -determinants according to a Pooled EGLS panel model (with cross-section weights)

 12

Source: Compiled by authors.

Table 13 : Sectoral industrial production in Japan -Determinants according to a Pooled EGLS panel model (cross-section SUR)

 13

Source: Compiled by authors.

Source: Compiled by authors.

CEAFJP DISCUSSION PAPER #18-03

for each of the variables. Hence, we proceed with the individual variables considering them as I(1) variables which have to be differenced once in order to become stationary. The results are available on request.

We first conduct sector-specific Autoregressive Distributed Lag (ARDL) estimations based on first differences of the (non-pooled) variables.

Sector-specific autoregressive distributed lag-estimations for Japanese industrial production

The pattern of the results of our ARDL estimations of eq. ( 2) based on first differences of the (non-pooled) sector-specific and two further non-sector specific variables are displayed in Table 7. We rely on the same mechanics of ARDL model selection which have already been explained in section 3.1.4. This time, entries in the "selected model" column indicate the model selected according the Akaike information criterion. The first entry in brackets denotes the lags of the endogenous lagged variable, the second entry corresponds with the lag number of the change of the sector-specific exchange rate, the third entry indicates the lag number of the change in US industrial production and the fourth stands for the lag number of the change in Japanese industrial input prices. Deviating from the specification of eq. ( 2) we leave out sectorspecific exports for reasons explained further above. Column 2 in Table 7 containing the Yen exchange rate effect denotes the significant estimated coefficients for the Yen exchange rate in the selected ARDL model. The total exchange rate effect (next column) is indicated in the third column of Table 7 by the sum of the period-specific significant entries. Values in brackets denote the respective significance level (probabilities). To illustrate, Table 8 contains a detailed ARDL estimation example for manufacturing sector as a whole. Its results are summarised in the first row of Table 7. Here, the impact coefficient of the exchange rate on the industrial production in the Japanese manufacturing sector is -0.61 with a rather high significance of below 0.01.

Table 7: Impact of the real effective Yen exchange rate on Japanese industrial production -A survey of estimated sector-specific ARDL models (excluding exports)

Source: Compiled by authors.

CEAFJP DISCUSSION PAPER #18-03

CEAFJP DISCUSSION PAPER #18-03

The ordinary Pearson correlations conducted for the example of the Japanese manufacturing sector displayed in Table 9 demonstrate that exports of the Japanese manufacturing sector dominate any impact on the industrial production of the same sector. Our additional robustness checks show that we can generalise this statement for all sectors considered here: branchspecific exports dominate any impact on the respective sector's industrial production.

As a robustness check, we estimated sector-specific ARDL models, this time including Japanese exports. The results are surveyed in Table 10. The pattern of the results does change in the sense that only for chemicals, electrical equipment, rubber, paper and food there is still a significant impact of the sector-specific Yen exchange rate with the "correct" sign. CEAFJP DISCUSSION PAPER #18-03

Our estimations based on Pooled EGLS (cross-section weights, Table 12) reveal that the (sector-specific) Yen exchange rate negatively impacts sectoral industrial production in the chemicals, the optical instruments, the paper, the rubber, the textiles and the wood sectors.

The expected negative sectoral exchange rate effect emerges from our Pooled EGLS (Cross-section SUR) estimates for the chemicals, the optical instruments, the rubber, the textiles, the transport equipment and the wood sectors (Table 13). We would like to stress that the evidence of negative Yen exchange rate impacts on Japanese sectoral industrial production is thus confirmed by the most sophisticated estimation method applied by us in this paper: the feasible GLS specification correcting for both cross-section heteroskedasticity and contemporaneous correlation (cross-section SUR).

Thus, for five industrial sectors (chemicals, optical instruments, rubber, wood and textiles) all three models find negative and significant coefficient estimates for the Yen exchange rate, while such effects are found for a further sector (paper) in two out of the three estimated estimation methods (Pooled Least Squares and Pooled EGLS/Cross-section SUR). Furthermore, Pooled EGLS finds significant negative effect also for the transport equipment sector. In all cases, the sectoral Yen exchange rate enters the final model consistently with a lag of three months (time-to-build effect). What is more, the selected empirical models are rather parsimonious in terms of the number of variables included. The (changes in the) Japanese industrial input price (INDINPUTPRICE) and US industrial production (INPUSA) are not part of the finally selected model. However, the lagged endogenous variable (change in Japanese industrial production) turns out to be highly significant throughout. Hence, in accordance with, for instance, [START_REF] Belke | Real Impacts of Intra-European Exchange Rate Variability: A Case for EMU?[END_REF], we feel legitimised to argue that our final empirical models can be interpreted as (at least partly) reduced forms against which we are able to separate the additional marginal impacts of the Yen exchange rate and Japanese exports on industrial production. 11 Finally, critics may argue that the empirical realisations of the R-squared are not overly high. However, this type of evidence is quite typical of regressions of changes on changes. What is more, the R-squared in our estimations appears to be located in the upper possible range.

Summing up, in our monthly data analysis with sector-specific data, both ARDL estimations and estimations of a mixed time series-cross section model suggest small but highly significant effects of the real Yen effective exchange rate (REER) on industrial production for a broad range of industrial sectors. 12 11. This interpretation is valid for all finally selected model specifications in this paper which do not include the full set of variables proposed by theory, i.e., by our equations ( 1) and (2). 12. In Section 3.2 we spared cointegration analysis because the time span 2000 to 2017 is much shorter than the one available in Section 3.1., i.e. 1970 to 2016. Note that using quarterly data (Section 3.2) instead of annual ones (Section 3.1) does not help, because integration and cointegration are sample (period) properties which are not necessarily dependent on the number of observations.

Appendix


Alexander Kolpakov

Alan W Reid

Leone Slavich

Leone Slavich Embedding

Embedding arithmetic hyperbolic manifolds

Introduction

Throughout the paper, all manifolds are assumed to be connected unless otherwise specifically stated. A complete, orientable, finite volume hyperbolic n-manifold M bounds geometrically if M is realized as the totally geodesic boundary of a complete, orientable, finite volume, hyperbolic (n + 1)-manifold W . There has been recent interest in a number of problems related to geometric boundaries (c.f. [START_REF] Kolpakov | Some hyperbolic three-manifolds that bound geometrically[END_REF], [START_REF] Kolpakov | Hyperbolic four-manifolds, colourings and mutations[END_REF], [START_REF] Long | Constructing hyperbolic manifolds which bound geometrically[END_REF], [START_REF] Slavich | A geometrically bounding hyperbolic link complement[END_REF] and [START_REF] Slavich | The complement of the figure-eight knot geometrically bounds[END_REF]).

One could ask for a weaker property, namely, that M simply embeds totally geodesically in a complete, orientable, finite volume, hyperbolic (n+1)-manifold W (which was considered in [START_REF] Martelli | Hyperbolic three-manifolds that embed geodesically[END_REF]). In this case cutting W along M either produces a single copy of M that bounds geometrically (if M happens to be separating in W ), or two copies of M that bound some hyperbolic (n + 1)manifold (otherwise).

In order to state the main result of this note, let us introduce the following notation. For a finitely generated group Γ we let Γ (2) = γ 2 : γ ∈ Γ . We also refer the reader to §2 for the other necessary definitions of terms in Theorem 1.1.

Theorem 1.1. Let M = H n /Γ (n ≥ 2) be an orientable arithmetic hyperbolic n-manifold of simplest type.

In many cases we are also able to prove that M bounds geometrically. Unlike the constructions of [START_REF] Kolpakov | Hyperbolic four-manifolds, colourings and mutations[END_REF], [START_REF] Slavich | A geometrically bounding hyperbolic link complement[END_REF] and [START_REF] Slavich | The complement of the figure-eight knot geometrically bounds[END_REF], our methods are algebraic and similar to [START_REF] Long | Constructing hyperbolic manifolds which bound geometrically[END_REF]. Indeed, Theorems 1.1 and 1.4 extend the results in [START_REF] Long | Constructing hyperbolic manifolds which bound geometrically[END_REF] which provide for all n examples of orientable arithmetic hyperbolic nmanifolds of simplest type that embed (in fact, bound geometrically) in an orientable arithmetic hyperbolic (n + 1)-manifold. The crucial new ingredient that we rely on is the recent work of Agol and Wise on proving separability, as applied in [START_REF] Bergeron | Hyperplane sections in arithmetic hyperbolic manifolds[END_REF] in the setting of arithmetic groups of simplest type. of finite index in Γ. By definition H(R k ) is an arithmetic subgroup of H, and evidently Γ ⊃ Γ ∩ H(R k ) ⊃ Γ ∩ G(R k ), and so Γ ∩ H(R k ) is an arithmetic subgroup of Γ.

Set Λ 1 = γ∈Γ γH(R k )γ -1 . Since Γ < G(k) < H(k), each element γ ∈ Γ commensurates H(R k ), and so by the previous paragraph Λ 1 is a finite intersection of H(k)-conjugates of H(R k ), which are all therefore commensurable arithmetic subgroups of H. Hence Λ 1 is an arithmetic subgroup of H, and moreover Λ 1 is normalized by Γ. It follows that Λ = Γ • Λ 1 is the required arithmetic subgroup of H contained in H(k).

There are many situations where all arithmetic lattices of G are contained in G(k). For example, it is a result of Borel [START_REF] Borel | Density and maximality of arithmetic subgroups[END_REF] (see also [START_REF] Allan | The problem of maximality of arithmetic groups, in: Algebraic Groups and Discontinuous Subgroups[END_REF]) that this takes place if G is a centreless linear algebraic group, and so this holds in particular if G = G.

However, this is often not the case, which we illustrate in the following example.

Example. Let Γ = SL(2, Z), and let n > 1 be a square-free integer. We can construct arithmetic groups Γ n < SL(2, R) commensurable with SL(2, Z) that are not subgroups of SL(2, Q).

Let Γ 0 (n) be the subgroup of SL(2, Z) consisting of matrices which are congruent to an upper triangular matrix modulo n. Then the element

τ n =   0 1/ √ n - √ n 0  
normalizes Γ 0 (n) and so the group Γ n = Γ 0 (n), τ n is an arithmetic subgroup of SL(2, R) commensurable with Γ and obviously not contained in SL(2, Q).

Using this example, we can easily show that a more general statement of Proposition 2.1 for embedding arithmetic groups in other arithmetic groups does not hold.

Example: Embed SL(2, R) < SL(3, R) as shown below: SL(2, R) | 0 0 | 1 ,
and consider the arithmetic subgroup

Γ n < SL(2, R) constructed above with Γ n → SL(3, R) embedded as Γ n | 0 0 | 1 .
Since SL(3, C) is centreless, it follows from the aforementioned result of Borel that this subgroup of SL(3, R) cannot be a subgroup of any arithmetic subgroup of SL(3, R) commensurable with SL(3, Z). However, it can also be seen directly as follows.

With τ n the involution as above, set

γ n = τ n | 0 0 | 1 .
If γ n ∈ Λ, an arithmetic group commensurable with SL(3, Z), then Λ contains a normal subgroup N which is a finite index subgroup of SL(3, Z). Hence N contains an element of the form:

x =   1 0 P 0 1 0 0 0 1   ,
for some P ∈ Z. Since N is a normal subgroup of Λ, we have that γ n xγ -1 n ∈ N . However,

γ n xγ -1 n =   1 0 0 0 1 -P √ n 0 0 1   ,
which is never an element of SL(3, Z).

Arithmetic subgroups of orthogonal groups

In this section we adapt some of the discussion of §2 to (special) orthogonal groups of quadratic forms.

Quadratic forms and arithmetic lattices

Let k be a totally real number field of degree d over Q equipped with a fixed embedding into R which we refer to as the identity embedding, and denote the ring of integers of k by R k . Let V be an (n + 1)-dimensional vector space over k equipped with a non-degenerate quadratic form f defined over k which has signature (n, 1) at the identity embedding, and signature (n + 1, 0) at the remaining d -1 embeddings.

Given this, the quadratic form f is equivalent over R to the quadratic form

x 2 0 + x 2 1 + • • • + x 2 n-1 -x 2 n
, and for any non-identity Galois embedding σ : k → R, the quadratic form f σ (obtained by applying σ to each entry of f) is equivalent over R to

x 2 0 + x 2 1 + • • • + x 2 n-1 + x 2 n .
We call such a quadratic form admissible.

Let F be the symmetric matrix associated with the quadratic form f and let O(f) (resp. SO(f)) denote the linear algebraic groups defined over k described as:

O(f) = {X ∈ GL(n + 1, C) : X t FX = F} and SO(f) = {X ∈ SL(n + 1, C) : X t FX = F}. For a subring L ⊂ C, we denote the L-points of O(f) (resp. SO(f)) by O(f, L) (resp. SO(f, L)). An arithmetic lattice in O(f) (resp. SO(f)) is a subgroup Γ < O(f) commensurable with O(f, R k ) (resp. SO(f, R k )).
Note that an arithmetic subgroup of SO(f) is an arithmetic subgroup of O(f), and an arithmetic subgroup Γ < O(f) determines an arithmetic subgroup Γ ∩ SO(f) in SO(f).

Apart from the case of n = 3, SO(f) is a connected absolutely almost simple semi-simple algebraic group defined over k. It is centreless when n is even, but has a non-trivial centre when n is odd.

Arithmetic groups of simplest type

Next we describe a particular construction of arithmetic subgroups of the Lie group Isom(H n ) via admissible quadratic forms.

Some notation

Let J n denote both the quadratic form

x 2 0 + x 2 1 + • • • + x 2 n-1 -x 2 n
, and the diagonal matrix associated with the form. We identify hyperbolic space H n with the upper half-sheet {x ∈ R n+1 :

J n = -1, x n > 0} of the hyperboloid {x ∈ R n+1 : J n = -1}, and letting O(n, 1) = {X ∈ GL(n + 1, R) : X t J n X = J n },
we can identify Isom(H n ) with the subgroup of O(n, 1) preserving the upper half-sheet of the hyperboloid {x ∈ R n+1 : J n = -1}, denoted by O + (n, 1).

Moreover O + (n, 1) is isomorphic to PO(n, 1) (the central quotient of O(n, 1)). With this notation, Isom + (H n ) = SO + (n, 1), the index 2 subgroup in O + (n, 1), which is the connected component of the identity of O(n, 1).

Note that when n is even, Isom(H n ) can be identified with SO(n, 1) by observing that there exists an isomorphism ϕ : O + (n, 1) → SO(n, 1) defined by ϕ(T ) = (detT) T, which restricts to the identity on SO + (n, 1).

Some arithmetic groups

To pass to arithmetic subgroups of O + (n, 1) and SO + (n, 1), we first note from §3.1 that, given an admissible quadratic form defined over k of signature (n, 1), there exists

T ∈ GL(n + 1, R) such that T -1 O(f, R)T = O(n, 1). A subgroup Γ < O + (n, 1) is called arithmetic of simplest type if Γ is commensurable with the image in O + (n, 1
) of an arithmetic subgroup of O(f) under the conjugation map above. An arithmetic hyperbolic n-manifold M = H n /Γ is called arithmetic of simplest type if Γ is. The same set-up using special orthogonal groups constructs orientation-preserving arithmetic groups of simplest type (and orientable arithmetic hyperbolic n-manifolds of simplest type).

We will also use the following notation. For a subring R ⊂ R, let O + (f, R) (resp. SO + (f, R)) be the subgroup of O(f, R) (resp. SO(f, R)) that leave both components of the cone {x ∈ R n+1 : f(x) < 0} invariant. Note that since T provides an equivalence of the form f with J n , it follows that f(x) = J n (T x), and

T -1 O + (f, R)T = O + (n, 1).
Notice that when n is even, arithmetic subgroups of simplest type in O + (n, 1) can be identified with the image of an arithmetic subgroup of SO(f) (using the discussion in §4.1). Moreover, since SO(f) is centreless, as remarked above, by [START_REF] Borel | Density and maximality of arithmetic subgroups[END_REF], all of its arithmetic subgroup are contained in SO(f, k). Therefore, when n is even any arithmetic group of simplest type in O + (n, 1) is constructed from an arithmetic subgroup of SO(f, k) for some admissible quadratic form f.

Locating arithmetic subgroups

In this subsection we prove the following result that will be an important step towards proving Theorem 1.1.

Proposition 4.1. Let f be an admissible quadratic form of signature (n, 1) defined over a totally real field k, g an admissible quadratic form of signature (n + 1, 1) defined over the same field k and suppose that G = SO(f) < H = SO(g). Let Γ be an arithmetic subgroup of G. Then:

1. If n is even, Γ embeds in an arithmetic subgroup of H.

2.

If n is odd, Γ (2) embeds in an arithmetic subgroup of H.

Proof. Part (1) follows from Proposition 2.1 and the previous remark that for n even, all arithmetic subgroups of G are contained in G(k). The second part follows from the proof of [10, Lemma 10] which shows that Γ (2) is a subgroup of G(k), and then we argue as in the even-dimensional case.

Indeed, both statements of Proposition 4.1 also apply to arithmetic subgroups of O + (f, R). This can be deduced from [10, Lemmas 6 and 10].

We record the following immediate corollary of Proposition 4.1 and the remark above.

Corollary 4.2. Let Γ be an arithmetic subgroup of O + (n, 1) of simplest type arising from an admissible quadratic form f of signature (n, 1) defined over a totally real field k. Suppose that there is an admissible quadratic form g of signature (n + 1, 1) defined over the same field k, with O(f) < O(g). Then:

1. If n is even, Γ embeds in an arithmetic subgroup of O + (n + 1, 1) of simplest type.
2. If n is odd, Γ (2) embeds in an arithmetic subgroup of O + (n + 1, 1) of simplest type.

The proof of Theorem 1.1 will follow from Corollary 4.2 once we have arranged the set-up of algebraic groups required by its statement, together with certain separability results. The former is done in §5, and the latter in §6 and §7.

Before that, we finish this section with some comments about adjoint groups in this context.

Comments on adjoint groups

We begin by discussing one of the issues on embedding arithmetic groups when n is odd. Thus, as before, let f be an admissible quadratic form of signature (n, 1) defined over a totally real field k and g an admissible quadratic form of signature (n + 1, 1) defined over k, as well. As above, let G = SO(f) < SO(g) = H. If n is even, then the inclusion map ι : G → H descends to the adjoint groups G → H. The reason for this is that G = G and so although the centre of H is non-trivial it is disjoint from G. However, when n is odd the inclusion map does not extend, precisely because in this case H = H and the central quotient is actually an isomorphism. In particular the centre of G is preserved; i.e. the restriction of the adjoint map H → H to G is not the adjoint map.

Regardless of n, the subgroup SO + (f, R) < SO(f, R) described in §4.2 is centreless, and so this does inject under the adjoint map. However, it is still not clear that one can arrange the image of an arithmetic group to lie in the k-points.

We now provide a number of examples to highlight some of the issues. The first example serves as a counterpoint to the discussion about SL(2, Z) given in §2.

Example. Let f(x, y, z) = xz -y 2 . Clearly f is an isotropic quadratic form of signature (2, 1). Indeed, in the notation above, SO + (f, Z) is isomorphic to PSL(2, Z) and it can be seen from [22, Section 2] that every maximal arithmetic subgroup commensurable with PSL(2, Z) is mapped isomorphically (by a homomorphism that is defined over Q) to a subgroup of SO + (f, Q). That is to say, these maximal groups can be realised as subgroups of the Q-points unlike the situation of the embedding of SL(2, Z) → SL(3, Z) described in §2.

We now consider a similar example when n = 3.

Example. Let π = 2 + i ⊂ O 1 = Z[i
] be one of the prime ideals in O 1 of norm 5. Following the set-up in §2, let Γ 0 (π) be the subgroup of SL(2, O 1 ) consisting of matrices which are congruent to an upper triangular matrix modulo π. Then the element

τ =   0 1/ √ 2 + i - √ 2 + i 0   normalizes Γ 0 (π)
and so the group Γ = Γ 0 (π), τ is an arithmetic subgroup of SL(2, C) commensurable with SL(2, O 1 ) and not contained in SL(2, Q(i)). For convenience, we will continue to use the same notation on passage to PSL(2, O 1 ).

Let f denote the quadratic form

x 0 x 1 +x 2 2 +x 2 3 . The homomorphism φ : PSL(2, C) → SO + (f, R)
given below:

a 0 + a 1 i b 0 + b 1 i c 0 + c 1 i d 0 + d 1 i -→     d 2 0 + d 2 1 -b 2 0 -b 2 1 b 0 d 0 + b 1 d 1 b 1 d 0 -b 0 d 1 -c 2 0 -c 2 1 a 2 0 + a 2 1 -a 0 c 0 -a 1 c 1 -a 1 c 0 + a 0 c 1 2(c 0 d 0 + c 1 d 1 ) -2(a 0 b 0 + a 1 b 1 ) 2(b 0 c 0 + a 1 d 1 ) + 1 2(b 1 c 0 -a 0 d 1 ) 2(c 0 d 1 -c 1 d 0 ) 2(a 1 b 0 -a 0 b 1 ) 2(b 1 c 0 -a 1 d 0 ) 2(a 0 d 0 -b 0 c 0 ) -1    
describes an isomorphism in which PSL(2, O 1 ) maps into SO + (f, Z). However notice that τ has image with (1, 2)-entry equal to -1

√ 2+i 2 = -1 √ 5
, and so the image of Γ is not contained in SO + (f, Q) either.

Note that, as is easily checked, Γ (2) < SO + (f, Q).

Remark 4.3. The isomorphism described above was worked out by Michelle Chu whilst correcting a claimed isomorphism in [9, p. 463] which turned out to be incorrect.

Embedding orthogonal groups

In this section we describe how to arrange the embedding of orthogonal groups required in §4.3.

Proposition 5.1. Let f be an admissible quadratic form of signature (n, 1) defined over a totally real field k. Then there is an admissible quadratic form g of signature (n + 1, 1) over k, so that O(f) < O(g) (and SO(f) < SO(g)).

Proof. To prove Proposition 5.1, we first reduce it to the case when f is a diagonal quadratic form, namely: Claim: Suppose that f is represented by the admissible diagonal quadratic form a 0 x 2 0 + a

1 x 2 1 + • • • + a n-1 x 2 n-1 -bx 2 n
, where a i ∈ R k are all positive and square free for i = 0, . . . , n -1, and b ∈ R k is positive and square free. Then there is an admissible diagonal quadratic form g of signature (n + 1, 1) with O(f) < O(g).

We defer the proof of the Claim and deduce Proposition 5.1. To that end, let f be an admissible quadratic form, so that standard properties of quadratic forms (see [START_REF] Lam | Algebraic Theory of Quadratic Forms[END_REF]) provide an equivalence over k of f to an admissible diagonal quadratic form f 0 . In particular, there exists T ∈ GL(n+1, k) so that T -1 O(f 0 )T = O(f). Assuming the Claim, and applying it to f 0 , we have an admissible diagonal quadratic form g 0 so that O(f 0 ) < O(g 0 ). We can extend T in a natural way to define a matrix

T = 1 | 0 0 | T ∈ GL(n + 2, k),
which provides an equivalence of the diagonal form g 0 to an admissible quadratic form g with O(f) < O(g).

We now prove the Claim. Let us start with some comments about the form f. If f is anisotropic over k then we can assume that b = a i for i = 0, . . . , n -1. Since O(λf) = O(f) for all λ ∈ k * , we can multiply f by a 0 -1 and therefore assume that a 0 = 1, and also that all other coefficients are square-free. Suppose first that k = Q. In this case we can take g = y 2 + f, which will be again a quadratic form over k. Then O(g, R k ) is cocompact, as follows from [START_REF] Morris | Introduction to Arithmetic Groups[END_REF]Proposition 6.4.4]. Now suppose k = Q. By Meyer's Theorem (see [31, §3.2, Corollary 2]), if n ≥ 4 (i.e. f is an indefinite quadratic form over Q in at least 5 variables) then f is isotropic and so taking g = y 2 +f is an admissible isotropic quadratic form. When n = 3, since by Meyer's theorem any admissible quadratic form of signature (4, 1) will be isotropic, we can simply take g = y 2 + f once again. In either case, O(g, Z) has finite co-volume [5, Theorem 7.8] but is not cocompact [START_REF] Morris | Introduction to Arithmetic Groups[END_REF]Proposition 5.3.4].

When n = 2, then we set g = q y 2 + f, where f is a ternary quadratic form of signature (2, 1) and q is a positive rational number. If f is isotropic, then so is g, and we can simply put q = 1. As above, in this case, the group O(g, Z) has finite co-volume but is not cocompact. If f is anisotropic, then there exists a positive rational number q such that -q is not represented by f over Q (see Lemma 10.1 of the Appendix). The form g = q y 2 + f is the required anisotropic quadratic form. The group O(g, Z) is cocompact by [START_REF] Morris | Introduction to Arithmetic Groups[END_REF]Proposition 5.3.4].

We can improve the embedding described in Proposition 5.1 in the following sense.

Corollary 5.2. In the notation established above, if R ⊂ R is a subring, then O + (f, R) can be embedded in SO + (g, R).

Proof. As above we write g = ay 2 + f for some a ∈ R k . Define a homomorphism ψ : O + (f, R) → SO(g, R) by Example:

M = ψ(M ) = det(M ) | 0 0 | M . We claim that M ∈ SO + (g, R). It is clear that g( M (v)) = g(v) for every v = (y, x 0 , . . . , x n ) ∈ R n+2 , since M ∈ O(f),
Let g = x 2 + y 2 + z 2 -7w 2
. This is an anisotropic form over Q due to the fact that 7 is not a sum of three integer squares (see for example [START_REF] Serre | A Course in Arithmetic[END_REF]Appendix,p. 45]), and a number is a sum of three integers if and only if it's a sum of three rational squares by the Davenport-Cassels lemma (see [START_REF] Serre | A Course in Arithmetic[END_REF]Lemma B,p. 46]). Thus, SO + (g , Z) defines a cocompact arithmetic Kleinian group H of simplest type.

Now we can embed a totally geodesic sub-2-orbifold into the quotient space H 3 /H as follows. First, consider the ternary form f = x 2 +z 2 -3w 2 . This form is also anisotropic over Q by reasons analogous to the above. Thus, it defines a cocompact arithmetic Fuchsian group G of simplest type. Set g = 21y 2 + f. We claim that the form g is equivalent to g over Q.

To see this, since the signatures of g and g coincide, it suffices to check their discriminants and Hasse invariants [START_REF] Serre | A Course in Arithmetic[END_REF]Theorem 9,p. 44]. We have that d(g) = -7 • 3 2 and d(g ) = -7. Also, we compute (g) = (21, -3) = (-(-3) • 7, -3) = (7, -3) = 1, by the standard Hilbert symbol's properties [START_REF] Serre | A Course in Arithmetic[END_REF]Proposition 2,p. 19]. Analogously, (g ) = (1, -7) = 1, by a straightforward check, over any Q p . Thus, we have an example of an arithmetic Kleinian group containing an arithmetic Fuchsian subgroup, whose embedding can be observed on the level of the corresponding quadratic forms as described at the end of the of the proof of Proposition 5.1.

Separability and consequences

To pass from embedding of subgroups to actual embeddings of manifolds we need to use recent progress on certain separability properties of these arithmetic groups of simplest type.

Separability

Definition 6.1. Let Γ be a finitely generated, discrete subgroup of O + (n, 1). We say that Γ is geometrically finite extended residually finite (or GFERF for short) if every geometrically finite subgroup H of Γ is separable in Γ (i.e. H is closed in the profinite topology on Γ).

We record the following from [START_REF] Bergeron | Hyperplane sections in arithmetic hyperbolic manifolds[END_REF]. Theorem 6.2. Let Γ < O + (n, 1) be an arithmetic group of simplest type. Then Γ is GFERF.

The proof of Theorem 6.2 follows from [3, Corollary 1.12] (see also the final remark of the paper, which deals with the non-compact case).

An important consequence of GFERF that we will make use of is the following. Throughout we will denote the profinite completion of a group G by G, and the closure of a subset X ⊂ G by Cl(X). Lemma 6.3. Let Γ < O + (n, 1) be an arithmetic group of simplest type, and let H < Γ be a geometrically finite subgroup. Then Cl(H) in Γ is isomorphic to H.

Proof. By the universal property of profinite completions, there is a natural continuous surjective homomorphism φ : H → Cl(H) which extends the inclusion H < Γ (notice that Cl(H) is a closed subgroup of a profinite group and therefore is also profinite).

We need to show that the map φ is injective. Note that φ is injective if and only if, given any finite index subgroup

H 1 < H, there is a finite index subgroup Γ 1 < Γ such that Γ 1 ∩ H < H 1 .
By standard properties, all of the finite index subgroups of H are also geometrically finite and therefore separable in Γ. This implies that indeed, for any finite index subgroup H 1 in H, there exists a finite index subgroup Γ 1 < Γ such that Γ 1 ∩ H = H 1 . For details, see [START_REF] Reid | Profinite properties of discrete groups[END_REF]Lemma 4.6] and the preceding discussion.

We will make use of the previous remarks and the following lemma ([25, Lemma 3.1]) in the proof of Theorem 6.5. Lemma 6.7. Suppose that G is a residually finite good group and H is a virtual retract of G. Then H is good.

Proof of Theorem 6.5. The proof of Theorem 6.2 in [START_REF] Bergeron | Hyperplane sections in arithmetic hyperbolic manifolds[END_REF] actually shows that if Γ < SO + (n, 1) is an arithmetic lattice of simplest type, then there exists a finite index subgroup Γ 1 < Γ so that Γ 1 is a subgroup of a right-angled Coxeter group C (see also [START_REF] Bergeron | Hyperplane sections in arithmetic hyperbolic manifolds[END_REF]Theorem 1.10]). Moreover, the nature of the construction in [START_REF] Bergeron | Hyperplane sections in arithmetic hyperbolic manifolds[END_REF] (see also [START_REF] Chesebro | Some virtually special hyperbolic 3-manifold groups[END_REF] and [START_REF] Haglund | Finite index subgroups of graph products[END_REF]) shows that Γ 1 sits as a quasi-convex subgroup of C. Since by [START_REF] Haglund | Finite index subgroups of graph products[END_REF] Coxeter groups virtually retract onto their quasi-convex subgroups, there exists a finite index subgroup C 1 < C together with a retraction homorphism r :

C 1 → Γ 1 .
The result will now follow from Lemma 6.7 once we establish that C is good (since as remarked goodness is preserved by commensurability). This is a consequence of the fact that any Coxeter group C is virtually special in the sense of [START_REF] Haglund | Special cube complexes[END_REF] (see in particular [START_REF] Haglund | Special cube complexes[END_REF]Proposition 3.10], and [START_REF] Haglund | Coxeter groups are virtually special[END_REF]). More precisely, [START_REF] Haglund | Special cube complexes[END_REF] provides a finite index subgroup C 2 < C with C 2 a quasi-convex subgroup of a RAAG A, and hence again by [START_REF] Haglund | Finite index subgroups of graph products[END_REF] (see also [START_REF] Haglund | Special cube complexes[END_REF]) a finite index subgroup A 2 admitting a retraction onto C 2 . The group A is good since it is a RAAG and so Lemma 6.7 applies to show that C 2 is good. Since goodness is preserved by commensurability, C is good as required.

The implication of goodness that we require is the following [28, Corollary 7.6]: Proposition 6.8. Suppose that Γ is a residually finite, good group of finite cohomological dimension over Z. Then Γ is torsion-free.

The fundamental groups of hyperbolic manifolds are well-known to be residually finite by Malcev's theorem, and any hyperbolic manifold is a K(π 1 (M ), 1) space. Hence, this implies that π 1 (M ) is of finite cohomological dimension over Z. Using the remarks above on good groups, and Proposition 6.8 we therefore have the following corollary: Corollary 6.9. Let M be a hyperbolic manifold with good fundamental group. Then π 1 (M ) is torsion-free.

7 Completing the proof of Theorem 1.1 Given Proposition 5.1 and Corollary 4.2, the following result will complete the proof of Theorem 1.1. The proof of Proposition 7.1 is given in a much more general context in [START_REF] Mcreynolds | Controlling manifold covers of orbifolds[END_REF], however, for completeness we give a proof in §7.1 adapted to the case at hand. Proposition 7.1. Let Γ be a torsion-free arithmetic lattice in O + (n, 1), of simplest type, and Λ an arithmetic lattice of simplest type in O + (n+1, 1) such that Γ < Λ. Then there is a torsion-free subgroup Λ 1 < Λ of finite index such that Γ < Λ 1 .

Deferring the proof of Proposition 7.1 for now, we complete the proof of Theorem 1.1. in Λ. Then, for every index m, we can find an element h m in [h] which belongs to the intersection of all H i with i ≤ m.

The closure Cl([h]

) is compact and therefore, up to passing to a convergent subsequence, we can suppose lim m→∞ h m = h ∈ Cl([h]). The limit h necessarily belongs to the closures of all the H i 's, and therefore to Cl(Γ), which is a contradiction because of (4). Now since Λ is a lattice, there exist only a finite number [h 1 ], . . . , [h n ] of conjugacy classes of non-trivial elements of finite order in Λ. To see this, first recall that being a lattice, Λ admits a convex finite sided fundamental polyhedron D. Now, up to conjugation, we can suppose that a non-trivial element of finite order h fixes a face F of D. Such a face F will be adjacent to only a finite number of other copies of D, say D 1 , . . . , D k , and the element h necessarily maps D to one of the D i 's, say D l . Moreover, no other element of Λ maps D to D l . Since there are only a finite number of choices for the face F and for the domain D l , the result follows.

For each such class [h i ], there exists a finite index subgroup F i < Λ which contains Γ and is disjoint from [h i ]. Clearly the intersection

Λ 1 = n i=1 F i (5)
is a finite index subgroup of Λ which contains Γ and is torsion-free. This completes the proof.

8 Proof of Theorem 1.4

Throughout this section f is an admissible quadratic form of signature (n, 1) and N = H n /H a non-orientable manifold double-covered by M = H n /Γ with H (and hence Γ) arithmetic of simplest type.

Theorem 1.4 will be a consequence of the following theorem on recalling from §4.2 that when n is even all arithmetic groups of simplest type arise from SO(f, k). Theorem 8.1. In the notation above, if H is conjugate into O + (f, k) then N bounds geometrically.

Proof. For convenience we identify H < O + (f, k), and so Γ < SO + (f, k). Using Corollary 4.2 we can find an arithmetic hyperbolic (n + 1)-orbifold X = H n+1 /Λ with H < Λ. As in the proof Theorem 1.1 we can use Proposition 7.1 to pass to a further finite-sheeted cover X 1 → X so that N embeds. Indeed, using Corollary 5.2 X 1 can be chosen to be orientable, and the proof is now completed by the well-known fact: Lemma 8.2 ( [START_REF] Long | Constructing hyperbolic manifolds which bound geometrically[END_REF]). Suppose that N is a codimension 1 non-orientable embedded totally geodesic submanifold of an orientable hyperbolic manifold X, and M is the orientation cover of N . Then M bounds geometrically.

Low dimensions

In this section we discuss bounding in the case of n = 3 and n = 2.

Bianchi groups

The statement of Corollary 1.5 will follow once we identify the groups PSL(2, O d ) as subgroups of SO(f, Z) for certain isotropic quaternary quadratic forms f, c.f. [START_REF] James | Fuchsian subgroups of Bianchi groups[END_REF]Theorem 2.1]. Thus, we avoid passing to a mod 2-cover. In order to arrange for bounding we take M to be a double cover of a non-orientable manifold N = H 3 /H with H as in Theorem 8.1.

Surfaces

The question as to which closed surfaces bound geometrically a hyperbolic 3-manifold has been of some interest. It follows from the work of Brooks [START_REF] Brooks | Circle packings and co-compact extensions of Kleinian groups[END_REF] that the set of surfaces that bound geometrically is a countable dense subset of the appropriate moduli space, but beyond a few examples (c.f. [START_REF] Zimmermann | Hurwitz groups and finite group actions on hyperbolic 3-manifolds[END_REF] and [START_REF] Zimmermann | A note on surfaces bounding hyperbolic 3-manifolds[END_REF]) little by way of explicit families of surfaces are known to bound geometrically. Indeed, [START_REF] Zimmermann | A note on surfaces bounding hyperbolic 3-manifolds[END_REF] is concerned with finding such explicit families where the surfaces embed totally geodesically. One consequence of Theorems 1.1 and 1.4 is the following. Corollary 9.1. Let Σ = H 2 /Γ be a closed arithmetic hyperbolic 2-manifold. Then Σ embeds as a totally geodesic surface in an arithmetic hyperbolic 3-manifold. If Σ double covers a nonorientable surface then Σ bounds geometrically.

Since the (2, 3, 7) triangle group is arithmetic, any Hurwitz surface is arithmetic. Hence as a special case of Corollary 9.1 we recover the following result of B. Zimmermann [START_REF] Zimmermann | A note on surfaces bounding hyperbolic 3-manifolds[END_REF].

Corollary 9.2. Let Σ be a Hurwitz surface. Then Σ embeds as a totally geodesic surface in an arithmetic hyperbolic 3-manifold. Corollary 9.2 applies in particular to the Klein quartic curve. However, by [START_REF] Singerman | Automorphisms of compact non-orientable Riemann surfaces[END_REF] the Klein quartic does not double cover a non-orientable surface and so we cannot deduce from Theorem 8.1 that the Klein quartic bounds. However, we can use the results of [START_REF] Singerman | Automorphisms of compact non-orientable Riemann surfaces[END_REF] to deduce the following. Corollary 9.3. There are infinitely many Hurwitz surfaces that bound geometrically.

Appendix

In this appendix, we prove the following statement: Lemma 10.1. Let f be an indefinite ternary anisotropic quadratic form defined over Q. Then there exist infinitely many rational numbers of either sign not represented by f over Q.

  and also det( M ) = 1. Moreover, since the transformation M ∈ O + (f, R) preserves the upper half-sheet of the n-dimensional hyperboloid {f(x 0 , . . . , x n ) = -1} ∩ {x n > 0}, its image M preserves the upper half-sheet of the (n + 1)-dimensional hyperboloid {g(y, x 0 , . . . , x n ) = -1} ∩ {x n > 0}.

Preliminaries on algebraic and arithmetic groupsWe recall some basic terminology about linear algebraic groups. We refer the reader to[START_REF] Platonov | Algebraic Groups and Number Theory[END_REF] for further details.Throughout this section G ⊂ GL(n, C) will be a semi-simple linear algebraic group defined over a totally real number field k (with ring of integers R k ). Associated with G is its adjoint algebraic group G, which is also defined over k, and the homomorphism π :G → G is a central k-isogeny (see [36, Section 2.6.1]).For U a subring of C we let G(U ) = G ∩ GL(n, U) denote the subgroup of U -points of G. By an arithmetic subgroup of G we mean a subgroup of G(R) commensurable with G(R k ). The proposition below is a useful starting point for what is to follow. Proposition 2.1. Let G and H be semi-simple linear algebraic groups defined over k with G ⊂ H. Let Γ < G(k) be an arithmetic subgroup. Then there exists an arithmetic subgroup Λ < H(k) with Γ < Λ.Proof. Since Γ is arithmetic it is commensurable with G(R k ) and so Γ ∩ G(R k ) is a subgroup

Remark 10.2. The argument above actually shows that if the coefficients of the form f are integers, so that also the determinant d is, then it is possible to find both positive and negative integers not represented by the form f.

Acknowledgements

The first and third authors were partially supported by the Italian FIRB project "Geometry and topology of low-dimensional manifolds", RBFR10GHHH. The first author wishes to thank William Jagy for his introduction into the vast body of literature on quadratic forms and related topics. The second author was supported in part by an NSF grant and The Wolfensohn Fund administered through the Institute for Advanced Study. He would also like to thank the I.A.S and M.S.R.I. for their hospitality whilst working on this project. He also wishes to thank Yves Benoist (Université Paris-Sud) and Gopal Prasad (University of Michigan at Ann Arbor) for helpful conversations and correspondence. The third author was supported by a grant from "Scuola di scienze di base Galileo Galilei", and wishes to thank the Department of Mathematics, Università di Pisa, for their hospitality whilst working on this project. He also wishes to thank Bruno Martelli and Stefano Riolo (Università di Pisa) for several helpful conversations. All three authors thank the I.C.T.P., Trieste, Italy, and the organizers of the "Advanced School on Geometric Group Theory and Low-Dimensional Topology: Recent Connections and Advances" held at the I.C.T.P. in June 2016, which allowed them to work on this project at an early stage.

Cohomological goodness

We now introduce a particular class of groups which have the nice property that their profinite completion is torsion-free.

Let G be a profinite group, and M a discrete G-module (i.e. an Abelian group equipped with the discrete topology on which G acts continuously). Let C n (G, M ) be the set of all continuous maps from G n to M . Define the coboundary operator ∂ : C n (G, M ) → C n+1 (G, M ) in the usual way to produce a cochain complex C * (G, M ). The cohomology groups of this complex are denoted by H q (G, M ) and are called the continuous cohomology groups of G with coefficients in M . Definition 6.4. Let Γ be a finitely generated group. We say that Γ is good if, for all q ≥ 0 and for every finite Γ-module M the natural homomorphism of cohomology groups

induced by the inclusion Γ → Γ is an isomorphism between the cohomology of Γ and the continuous cohomology of Γ.

In general, establishing whether a group is good or not, is difficult. However, in our context we have the following (which is proved below). Theorem 6.5. Let Γ < O + (n, 1) be an arithmetic group of simplest type. Then Γ is good.

Indeed this follows for all finite volume hyperbolic 3-manifolds by the virtual fibering results of Agol [START_REF] Agol | The virtual Haken conjecture, with an appendix by I[END_REF] and Wise [START_REF] Wise | The structure of groups with quasiconvex hierarchy[END_REF], together with the following criterion due to Serre [32, Chapter 1, Section 2.6]: Proposition 6.6. Let Γ be a finitely generated group. Suppose that there exists a short exact sequence

where H and N are good groups, N is finitely generated and the cohomology group H q (N, M ) is finite for every q and every finite Γ-module M . Then Γ is good.

Moreover, goodness is preserved by commensurability and free products of good groups are good. Furthermore, controlled amalgams of good groups are good [START_REF] Grunewald | Cohomological goodness and the profinite completion of Bianchi groups[END_REF]. In particular (see e.g. [START_REF] Reid | Profinite properties of discrete groups[END_REF]Theorem 7.3]), it can be shown that the following groups are good:

1. finitely generated Fuchsian groups, 2. fundamental groups of compact 3-manifolds, 3. fully residually free groups, 4. Right Angled Artin groups (RAAG's), Proof of Theorem 1.1. Let M = H n /Γ be an orientable arithmetic hyperbolic n-manifold of simplest type. We will assume that n is even. The odd-dimensional case is handled exactly the same way replacing Γ by Γ (2) . By Corollary 4.2, there exists exists an arithmetic lattice Λ of simplest type in SO + (n + 1, 1) such that Γ < Λ. By Proposition 7.1, we can find a torsion-free subgroup Λ 1 < Λ with Γ < Λ 1 . Now by Theorem 6.2, Λ 1 is GFERF, and a standard consequence of this (see [START_REF] Scott | Subgroups of surface groups are almost geometric[END_REF]) is that M embeds in a finite sheeted cover of H n+1 /Λ 1 .

The final sentence in the statement of Theorem 1.1 follows from the proof of the Claim in Proposition 5.1 and the fact that closed arithmetic manifolds of simplest type are associated with quadratic forms either over a finite extension k of Q, k = Q, or with quadratic forms over Q which are anisotropic.

As we have seen above, isotropic quadratic forms over Q give rise to necessarily non-compact arithmetic manifolds. This completes the proof.

Proof of Proposition 7.1

In what follows Γ will always denote an arithmetic lattice of simplest type in O + (n, 1). Lemma 7.2. Suppose that η ∈ Γ has finite order, and [η] denotes the conjugacy class of η in Γ. Then Cl([η]) ⊂ Γ consists entirely of elements of finite order.

Proof. Let λ be an element of Cl([η]). Then there exists a convergent sequence {η j } ∞ j=0 of elements of [η] whose limit is λ. For each η j , by definition there exists an element β j ∈ Γ such that β -1 j η j β j = η. The sequence {β j } ∞ j=0 is a sequence in the compact group Γ and therefore admits a convergent subsequence {β l } ∞ l=0 with limit β. By continuity of taking inverses, we see that also {β -1 l } ∞ l=0 converges to β -1 . Finally:

Therefore η and λ are conjugate in Γ, and since η has finite order so does λ.

We now commence with the proof of Proposition 7.1.

Proof. Let Γ < Λ be as in the statement of Proposition 7.1, and h ∈ Λ be a non-trivial element of finite order. By Lemma 7.2, the closure of the conjugacy class of h in Λ consists entirely of elements of finite order. By Lemma 6.3 and Corollary 6.9, the closure of Γ in Λ is torsion-free. Therefore:

From this we can conclude that there is a finite-index subgroup H of Λ which contains Γ and is disjoint from [h]. To see this, suppose to the contrary, and fix a total ordering {H 1 , H 2 , . . .} on the finite-index subgroups of Λ which contain Γ. We have that Γ = ∞ i=1 H i , since Γ is separable

Proof. The Hasse-Minkowski theorem [START_REF] Lam | Algebraic Theory of Quadratic Forms[END_REF]Chapter 6.3] implies that f does not represent (over Q) a rational number q if and only if f does not represent q over a p-adic completion

We first deal with the case q = 0. By [31, Theorem 6, p. 36], a ternary form f does not represent 0 over Q p if and only if

where d(f) is the discriminant of f, (•, •) denotes the Hilbert symbol, and (f) is the Hasse invariant of the form f. Now suppose q = 0. By the Corollary after [31, Theorem 6, p.36] a ternary form f does not represent q over the Q p if and only if the following two conditions are satisfied:

Since f is anisotropic, the latter condition (8) (which does not depend on the choice of q) is satisfied in some Q p because of [START_REF] Brooks | Circle packings and co-compact extensions of Kleinian groups[END_REF]. Also, since f is indefinite, we have that p = ∞.

It remains to show that, given a prime number p, we can always find both a positive and a negative rational number q such that the former condition ( 7) is satisfied.

First, let us show that one can realise both positive and negative integers as squares in Q p . Let n be any integer which is co-prime with p. Such a number n is a p-adic unit in Z p , i.e. an invertible element of the ring of p-adic integers. Suppose p = 2 first. By [31, Theorem 3, p. 17 ], n as above is a square in Q p if and only if its image in U/U 1 = F * p is a square. Here, U = Z * p is the group of p-adic units, U 1 = 1 + p Z p and F p = Z/pZ. However, the latter is simply the first term in the p-adic expansion of n. It is therefore sufficient to choose n such that it is a quadratic residue in F * p , and there are clearly both positive and negative choices for such an integer n.

If p = 2, by [START_REF] Serre | A Course in Arithmetic[END_REF]Theorem 4,p. 18] in order for a 2-adic unit n to be a square it is necessary and sufficient that n = 1 (mod 8) or, equivalently, that the third term in the 2-adic expansion of r is 1. Once more, we can clearly pick n to be either positive or negative. This proves that for every prime number p there are infinitely many integers of either sign, which are realised as squares in Q p .

Finally, we choose a rational number q (positive or negative) such that -q/d is a square in Q p so that (7) is satisfied. Therefore f does not represent q over Q p and, by Hasse-Minkowski, over Q.