Yvette Vaguet

Oil and Gas towns in Western Siberia: past, present and future challenges

Introduction

Western Siberia, lying mainly in a permafrost area, has been urbanised for two decades or thereabouts. Russia has had the most notable experience of urbanisation within the circumpolar zone and the landscape bears the most consummate signs of this experience in the region.

Oil extraction constitutes the major determinant of this urbanisation. The Ob' region is one of the biggest oil and gas basins in the world, and contributes up to 68 % of the national production of oil and 91 % of gas. Thanks to Western Siberia, Russia is the first gas producer in the world, and the second for oil. This exploitation started in 1964. During the Soviet era, the development of the oil/gas industry came with the spread of modern society northwards within the almost uninhabited taiga first, and the tundra later, as far as the Yamal peninsula along the coast of the Arctic Ocean. When the USSR collapsed less than 30 years later, 23 out of 28 towns had been created ex-nihilo and were associated with considerable in-migration. Building human residential and industrial infrastructure on this scale and at this pace within such a harsh natural environment, almost without any permanent human settlement nearby, was a challenge in itself. Town and country planning currently have new challenges to face.

One can wonder if these urban centres were actual towns rather than just short-term settlements. The current debate concerns the future of these towns: whether they can survive beyond the peak oil or even earlier by flooding of the marshy Ob' basin due to global warming and its associated melting permafrost. Howev-er, what lies in the future could also be a flourishing economy with the prospect of oil-and-gas development northwards to the Arctic Ocean. Awaiting a response, the region still presents a positive migration balance and nowadays Western Siberian towns are trying to diversify their economic profile. Municipalities are also trying to improve the quality of life in order to retain, if not attract, inhabitants, especially the more highly skilled. Moreover, here and there in these towns, there are evident signs of gentrification, with the spread of the detached house model, as in many parts of the world.

To sum up, upcoming challenges question the resilience of this urban geographical system. The answer is not as easy as it might appear. This article introduces the complexity of the subject. Because of its unique 50-year experience, Western Siberia is a key area when considering urban development in the Arctic.

Past challenges: Building an urban system on the edge

The study area in this article covers three constituent entities of the Russian Federation. It includes two producing territories, namely the Autonomous Okrug of Khanty-Mansi (mostly oil producing), and towards the north, the Autonomous Okrug of Yamalo-Nenets along the Arctic Ocean coastline (mostly gas producing). It also includes the southern Oblast of Tyumen, which is not hydrocarbon producing itself but has administrative functions (fig. 1). The whole study region is 70 % the size of Greenland. The development of industry started in 1964 at the Samatlor oil field near the future town of Surgut (61°15'N). This was the beginning of a fantastic human adventure, which was also, one should note, an environmental disaster. From the start, modern society, both urban and industrial, was to be found in the south, mostly along the Trans-Siberian Railway. Meanwhile, the two huge northern Okrugs were the territory of indigenous peoples (the Khanty, Mansi and Nenets). Their main sub-sistence came from hunting, fishing, gathering and reindeer herding, as it still does today. Their territory was structured by rivers. There was neither major road nor airport, and their administrative centres (Khanty-Mansiysk (61°00'N) and Salekhard (66°32'N)) resembled fishing villages.

This economic transition brought not only industrialisation and urbanisation but also a rotation of the main regional strategic axis: from the east-west Trans-Siberian, it developed northwards towards the Arctic. New hydrocarbon fields progressively opened up in these environments, which were increasingly difficult for humans and human activities. In the mid-1960s, the first oil pipelines were laid in the middle of the Ob River in marshy areas, in the discontinuous permafrost and an active 80 cm thick layer. In the late 1980s, the Urengoi gas basin opened up in an area where permafrost is up to 400 m thick and contains up to 50 % of ice. It produces more gas than all the European countries put together. Its associated town, Novy-Urengoi (66°05'N), is one of thebiggest cities within the polar region (pop.≈100 000). Northwards, more recent gas fields extend across the continuous permafrost area, up to 500 m thick and containing up to 90 % of ice. Since 2000, a gas pipeline has connected the Yamal peninsula to Western Europe.

In-migration was high and directed towards the two northern Okrugs. The regional population increased from 1.1 million in 1959 to 5.2 million in 2002. Inmigration reduced the gap between North and South in many ways. First, numerically speaking, the picture of an empty Okrug was no longer a reality. If 83 % of Western Siberia's population were to be found in the southern Oblast of Tyumen in 1959, the northern two Okrugs had 57 % of the regional population by 1989. Secondly, in qualitative terms, the image of an urban and industrial Russian society in the southern part alongside a nomadic and rural indigenous society in the northern part is no longer true either [START_REF] Zaitseva | Population of Tyumen region in XXs[END_REF]Vaguet-Marchand Yv., 2004;[START_REF] Vaguet | Les Hydrocarbures, les villes et les hommes dans le Nord-Ouest Sibérien[END_REF]. Out of the 28 towns, 23 are new, created for the needs of the economic transition. Among those, 21 were created within the northern two Okrugs that registered the biggest urban increases within the country for half a century (fig. 2). Their population became urban (>77 %). Moreover, the new population came from all over the USSR and was unrelated to indigenous peoples, their lifestyle and arctic environment. Meanwhile, a complex transport network emerged with a major north-south railway, a main north-south road and airports in every town. Apart from towns, there are other kinds of settlements that were introduced by the economic transition. In order to get an official town status, an urban centre needs to apply for this status from the government when its population is greater than 12,000 inhabitants. While there are 28 urban centres with town status, there are also about 40 urban settlements without this official status. They have all arisen since the mid-1960s for the needs of the economic transition; they are gas plant or road maintenance facility centres, etc. Besides these, long distance commuter settlements should also be mentioned. They are small outposts in the taiga or tundra where oil and gas shift-workers come to work for a month or so, before going back home. Workers arrive mainly by air in the nearest town and the company picks them up at the airport by bus and drives them straight to the base. Thus, the town of Noyabrsk (63°12'N) for instance, had up to 10 % of its population employed as shift-workers, which is to say about 10,000 in 1991.

In this process of industrialisation and urbanisation of Western Siberia, the creation of a settlement follows a model. First, prospectors arrive and if hydrocarbons are found, human settlement starts. Newcomers are mostly young men (20-30 years old) and possessing few skills. They build all the required human infrastruc-ture (roads, pipelines, industrial plants, houses, etc.). Shortly afterwards, women arrive and the settlement registers a baby boom. Meanwhile the building stage is still ongoing. Finally, skilled and white collar workers arrive to run the industry. During this process, priority is given to the industrial production. Social infrastructure is always under-developed and the need for housing prevalent. Thus, there is never an end to the housing construction process. Finally, the increase in the population slows down owing to the natural growth rate rather than the migration balance.

The population of Western Siberia is consequently young (median: 32.3 years). Newcomers are mostly single, from 20 to 30 years old, and childless. There is mostly an absence of elderly people except in the case of family reunifications. Men are over-represented (up to 113 men to 100 women). This fact is common in all frontier areas around the world, but this particularity occurs here in a country that has the lowest sex ratio in the world (88 men to 100 women). Nowadays, this demographic increase is still high (+10 % between 1989 and 2002). This results from a natural increase, among the highest in the country, as well as from a migration balance which is still positive. It is unique in Northern Russia, which registered dramatic out-migration during the 1990s.

The city of Noyabrsk is a good example of this process. In 1973, oil flows and the first pioneers arrived. To start off with, people and material arrived by helicopter. The population suddenly soared when, a year later, the first train arrived at Noyabrsk from Surgut in the south. The latter city was the base town during the emergence of the new Noyabrsk settlement. Surgut was already an established settlement and provided the logistics for the new northward settlement. Every year, more than 15,000 people arrived in Noyabrsk. The rate of population increase culminated in 1977-78 with a mighty +224 %. The baby boom appeared in the 1980s with a maximum in 1987. Noyabrsk obtained the status of a city in 1982. Nowadays, the town maintains a population of about 100,000 residents, and the annual increase is due mostly to the natural growth rate. The town's population presents an average age of 26 years and elderly people are almost absent. On the 1999 age pyramid, the limited number of inhabitants aged 15-20 years marks the time of the first arrivals without children. The wide base of this pyramid marks the baby boom period (fig. 3). [START_REF] Engel | Public space in the 'bule cities' of Russia», 285-300, in STANILOV K. The post-socialst city[END_REF][START_REF] Gavrilova | Le complexe des villes pétrolières et gazières de Sibérie occidentale et sa mise en oeuvre (1960-1980[END_REF]Radvanyi J., 2000).

A new identity of Western Siberia emerged in this transition period. Economically, the region was far behind central Russia, like the whole of Asian Russia. Nowadays, the region resembles a "Siberian emirate" ranking in the top ten richest provinces of the Russian Federation, often just below the central regions (Moscow and St Petersburg) according to economic indicators (Direct Foreign Investment, Regional Gross Product). Although the high cost of living greatly outweighs high salaries, and the lack of housing leads to further disillusionment, the region is still attractive. It has gained an image of a rich, dynamic territory with real economic forces and young people showing entrepreneurship. However, the Soviet policy of regional planning has now been over for decades and all these towns face new challenges for the future.

Present and future challenges: Becoming a sustainable town

Filling up the tank of your car at a petrol station would cost far more than it would had the USSR not conducted this incredible challenge of populating the Arctic and Subarctic Western Siberia. The urban system of Western Siberia might therefore not have been in a market economy. Some settlements were still sustaining rapid growth when the USSR collapsed and the growth process stopped at this point. The Soviet government encouraged the development of small cities all over the national territory and hundreds of cities were created. Nowadays, the Russian government no longer supports territorial equity and goes rather for supporting the development of regional metropolitan centres. The decade of the 1990s presented a multi-faceted crisis (political, economic, demographic and social) that had a greater impact in the northern part of the country. People were more exposed to the crisis when they lived in a single industry town in the remote north. Many settlements emptied as people went back to the central parts of Russia. Thanks partially to huge oil and gas consumption in the world, Western Siberia is well known as an exception within circumpolar Russia. Its 2002 census shows a healthy demographic situation, increasing its population both by natural growth and a migration balance. However, one of the main challenges for the future facing these new towns is sustaining themselves beyond the peak oil.

The price of a barrel of oil has to remain high in order to cover the costs of extraction in such a harsh environment ($14/barrel compared to $4 in Kuwait and The United States). Every year, depending on the production, reserves and the price of oil, these towns are threatened with closure. Looking at both hydrocarbon production and population over time illustrates how much town and industry are linked together (fig. 4). Noyabrsk is associated with an oil field estimated to last 50 years and it has been exploited for 40 years now.

Nevertheless, oil and gas will continue to support Reserves are vast especially over the continental shelf (more than 18,000 billion m 3 of gas for the Siberian part). Nowadays, the oil and gas sector contributes up to 23 % GDP in the circumpolar zone. As producers, the combined two Okrugs of Western Siberia and Alaska are among the top four strongest circumpolar economies. Together with the Republic of Sakha (Russia), these four regions provide 60 % GDP in the circumpolar zone (AMAP, 2007). Towns in Western Siberia would benefit from ongoing developments of the industry, moving northwards into offshore extraction and transportation. Already, there is a very large-scale project in view for the Yamal peninsula. It includes building a gas processing plant, a seaport and an international airport. The new port is planned for 2018. Russia's Minister of Transport, Maxim Sokolov, says: "For Russia, this project will […] change the very position of Russia in the world's economy. This project will make Russia one of the leaders in developing the natural riches of the Arctic region." (The Voice of Russia, 2012). Global warming could help in this way especially with increasing shipping in the Arctic.

In a less optimistic point of view, global warming could lead to higher frequency of icebergs, a danger for extraction and shipping activities. Also, it could be responsible for sea and river flooding. It could also lead to the melting of permafrost, especially the newest, thinnest and most discontinuous parts that coincide with the oldest human installations. Therefore, oil and gas fields and towns of the low, marshy Ob' basin could end up under sea level or river water or otherwise could disappear due to subsidence from the effect of permafrost melting. Some authors have already registered several indicators of climate change in this region [START_REF] Grebenuk | New Challenge of the Modern Civilization and Gobal Economy in the North of the Western Siberia[END_REF]. Human activity has to cope with a fast changing environment.

The diversification of the local economy is another challenge. Indeed, Western Siberian towns, like many Soviet cities, are mainly mono-industrial and linked to the military sector [START_REF] Lyubovnyi | Macroeconomic Factors of Grouwth of Russian Cities and Town Planning Policy Objectives[END_REF]. Worse, some towns are mainly companyowned, therefore raising the question of the definition of a town. All 23 new towns were created ex-nihilo in response to the needs of the industry. Very few of them are not directly oil or gas producing centres. All of them struggle with economic diversification and see hardly any positive results especially for the most recent towns close to the Arctic Circle. For instance, Novy-Urengoi was created 35 years ago and has a population nowadays of approximately 100,000 inhabitants. Gazprom, the biggest gas company in the world, employs up to 70 % of its population. Thus Novy-Urengoi is often called a Gazprom-town. Its residents have the advantage of living off and being employed by the first Russian company. The company is more or less in charge of many key sectors such as health, education, sport, transport, and the municipality authority is hardly independent. The economic market tends to push the company into reducing its support for the social aspects of the city's life and to concentrate its activity on extraction. However, the municipality has neither the budget nor the knowledge to tackle transport issues, for instance, such as road maintenance, the tremendous increase in cars, or the strange new forest of electric cables connecting indoor power sockets in flats to car batteries at the base of buildings.

Southwards, Surgut, on mid-river Ob, is a good example of the first generation of these towns. Created for the oil industry in the 1960s, the town still depends mainly on this sector. Out of its 320,000 inhabitants, 100,000 are employed in this sector and Surgutneftegaz employs 80,000 of them. The company largely supports local infrastructure such as the stadium as well as health and cultural centres. Considering its geographical position, at the southern part of the permafrost and along the Ob River, river flooding and permafrost melting threaten Surgut. As one of the oldest oil centres, the aging of industrial installations is also a matter of concern.

Between Novy-Urengoi and Surgut, Noyabrsk is a third example showing a typical intermediate case among these towns. Noyabrsk is two decades younger than Surgut and boasts 100,000 inhabitants. The town produces both oil and gas. The industrial installations are getting old here too where the climate is harsher. Moreover, the municipality is facing the aging and the depressed state of its population. People rarely settle for long. Out-migration has always been high. A survey conducted during Soviet times showed that residents of Yamalo-Nenets Okrug where Noyabrsk lies, tended to stay fewer than 5 years in the same place [START_REF] Logunov | Demographic and social consequences of oil and gas development in Siberia[END_REF]. In the survey conducted in 2003, the wish to leave was ever present among adults and town officials note that the highly educated classes leave (survey Vaguet Yv. 2003; Noyabrsk official web site). The urban population is not indigenous. Therefore it is not well adapted to the local Arctic environment and has no affective links with it. People are always ready to move on, looking for a better salary and living conditions. The regular announcements of the forthcoming closure of the urban centre by the municipality itself contribute to maintaining local pessimism.

However, across Western Siberia, as in Russia, if a municipality finds it difficult to conduct an urban plan, the urban landscape changes as the result of the inhabitants' response to the new world order. The retail sec-tor is booming all over Russia, as well as in the Arctic region which goes to show that residents have become producers of their own urban space and are taking part in the economic diversification process. Concerning the residential function, the social mix of Soviet times is less and less the rule. Here and there, signs of gentrification can be noted. Some Soviet buildings have been renovated, with private access and a guard at the entrance. Looking at both satellite images and on the ground, urbanisation is visibly spreading. At the town's periphery, the detached house model is spreading as it is in many parts of the world. The boom in cottages shows the upper class is ready to invest here. In addition to the points mentioned above, it is important to bear in mind that young people feel at home and do not want to leave. They were born in these towns, which are now as old as them. If their parents play with the idea of going back south one day, the teenagers questioned do not share this desire (survey Vaguet Yv., 2003).

All the above challenges are of equal importance when questioning the ability of these towns to remain alive. It is now a question of inventing a sustainable town in an extreme environment. From this study, it is obvious that a young, talented, adaptable and entrepreneurial population is present in this region.

Conclusion

Western Siberia is exceptional, being one of the most important regions for oil and the biggest for gas production in the world system. There are about 30 towns and 2/3 of them were created in connection with the industry during Soviet times at a rapid pace in an extremely harsh environment for humans and human installations. This urban network would probably not have been created in Western Siberia if it had not been a communist country. However, these towns still attract workers and nobody can be sure that they will close one day. Global warming could be a positive factor locally though it could also announce natural and human disasters.

Because of its fifty-year experience, Western Siberia is a key area when considering urban development in the Arctic. The single-industry-, and sometimes the single-company-, economy is an often-shared characteristic of the (sub) Arctic urban centres. In addition, this trait is often an industry based on a non-renewable resource. The cost of extraction is always higher than at lower latitudes. The question of profitability is therefore more acute here. The diversification of the local economy can be difficult considering that it would imply fewer links between the major employer and the municipality. What is important is to invent now and tomorrow a long-term town in an extreme environment. Once again, humans have to face a challenge in Western Siberia. The right demographic blend is an asset for these towns. Residents are tackling the debate: they are inventing new functions, transforming Soviet urban landscapes and creating new urban spaces on the periphery.

Figure 1 :

 1 Figure 1: Western Siberia, its oil and gas towns

Figure 2 :

 2 Figure 2: The urban transition (database: SiberCities ©Yv. Vaguet)

Figure 3 :

 3 Figure 3: Age pyramid of Noyabrsk (database: SiberCities ©Yv. Vaguet)

Figure 4 :

 4 Figure 4: Population and Oil/Gas Production in Noyabrsk & Muravlenko(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001) (database: SiberCities ©Yv.Vaguet)

Figure 5 :

 5 Figure 5: The changing urban space (photographs: Y. Vaguet & A. Gaye)


Inkang Kim

Pierre Pansu

LOCAL RIGIDITY IN QUATERNIONIC HYPERBOLIC SPACE

In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.

1. Introduction 1.1. 4-dimensional lattices. Lattices in Sp(n, 1), n ≥ 2, when mapped to Sp(m, 1), cannot be deformed. This follows from K. Corlette's archimedean superrigidity theorem, [START_REF] Corlette | Archimedean superrigidity and hyperbolic geometry[END_REF]. What about lattices in Sp(1, 1), i.e. in 4-dimensional hyberbolic space ?

In this note we prove local rigidity of uniform lattices of Sp(1, 1) when mapped to Sp(2, 1). In complex hyperbolic geometry, such rigidity results were first discovered by D. Toledo, [START_REF] Toledo | Representations of surface groups on complex hyperbolic space[END_REF]. In [START_REF] Goldman | Representations of fundamental groups of surfaces[END_REF][START_REF] Goldman | Local rigidity of discrete groups acting on complex hyperbolic space[END_REF], W. Goldman and J. Millson gave a local explanation of this phenomenon. Our main result is an exact quaternionic analogue of theirs.

Start with a uniform lattice Γ in Sp(1, 1). There is an easy manner to deform the embedding ρ 0 : Γ → Sp(1, 1) → Sp(2, 1). Indeed, since Sp(2, 1) contains Sp(1, 1) × Sp [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF], it also contains many copies of Sp(1, 1) × U [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. If H 1 (Γ, R) = 0, which happens sometimes (see [START_REF] Millson | On the first Betti number of a constant negatively curved manifold[END_REF]), the trivial representation Γ → U(1) can be continuously deformed to a nontrivial representation ρ 1 . All such representations give rise to actions on quaternionic hyperbolic plane which stabilize a quaternionic line. Therefore, only deformations which do not stabilize any quaternionic line should be of interest.

Theorem 1.1. Let Γ ⊂ Sp(1, 1) be a lattice. Embed Γ into Sp(2, 1) as a subgroup which stabilizes a quaternionic line. If Γ is uniform in Sp(1, 1), then every small deformation of Γ in Sp(2, 1) again stabilizes a quaternionic line.

If Γ is non uniform in Sp(1, 1), then every small deformation of Γ in Sp(2, 1) preserving parabolics again stabilizes a quaternionic line.

Toledo's theorem inaugurated a series of global rigidity results by A. Domic, D. Toledo, [START_REF] Domic | The Gromov norm of the Kaehler class of symmetric domains[END_REF], K. Corlette, [START_REF] Corlette | Flat G-bundles with canonical metrics[END_REF], M. Burger, A. Iozzi and A. Wienhard, [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF]. By global rigidity, we mean the following : a certain characteristic number of representations, known as Toledo invariant, is maximal if and only if the representation stabilizes a totally geodesic complex hypersurface. It is highly expected that such a global rigidity should hold in quaternionic hyperbolic spaces, but we have been unable to prove it. Note that since Sp(1, 1) = Spin(4, 1) 0 , there exist uniform lattices in Sp(1, 1) which are isomorphic to Zariski dense subgroups of Sp(4, 1), see section 7.

Question. Let Γ ⊂ Sp(1, 1) be a uniform lattice. Embed Γ into Sp [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. Can one deform Γ to a Zariski dense subgroup? 1.2. 3-dimensional lattices. Uniform lattices in 3-dimensional real hyperbolic space can sometimes be deformed nontrivially in 4-dimensional real hyperbolic space, see [START_REF] Thurston | The geometry and topology of 3-manifolds[END_REF], chapter 6, or [START_REF] Apanasov | Bending and stamping deformations of hyperbolic manifolds[END_REF]. Nevertheless, when they act on quaternionic plane, all small deformations stabilize a quaternionic line, although the action on this line can be deformed non trivially.

Theorem 1.2. Let Γ ⊂ Spin(3, 1) 0 be a lattice. Embed Spin(3, 1) 0 into Spin(4, 1) 0 = Sp(1, 1) and then into Sp(2, 1) in the obvious manner. This produces a discrete subgroup of Sp(2, 1) stabilizing a quaternionic line.

If Γ is uniform in Spin(3, 1) 0 , then every small deformation of Γ in Sp(2, 1) again stabilizes a quaternionic line.

If Γ is non uniform in Spin(3, 1) 0 , then every small deformation of Γ preserving parabolics again stabilizes a quaternionic line.

If the assumption on parabolics is removed, nonuniform lattices in Spin(3, 1) 0 can be deformed within Spin(3, 1) 0 , see [START_REF] Thurston | The geometry and topology of 3-manifolds[END_REF], chapter 5.

Question. Let Γ be a non uniform lattice in Spin(3, 1) 0 . Map it to Sp(2, 1) via Spin(4, 1) 0 = Sp [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. Can one deform Γ to a Zariskidense subgroup? 1.3. 2-dimensional lattices. Uniform lattices in real hyperbolic plane, when mapped to SU(2, 1) using the embedding SO(2, 1) → SU(2, 1), can be deformed to discrete Zariski-dense subgroups of SU(2, 1). On the other hand, lattices mapped via SU(1, 1) and SU(2, 1) are more rigid, as shown by W. Goldman and J. Millson,[START_REF] Goldman | Local rigidity of discrete groups acting on complex hyperbolic space[END_REF]. This fact has been recently extended to higher rank groups by M. Burger, A. Iozzi and A. Wienhard, [START_REF] Burger | Surface group representations with maximal Toledo invariant[END_REF].

It turns out that this form of rigidity of surface groups does not apply to the group Sp(2, 1). Theorem 1.3. Let Γ be the fundamental group of a closed surface of genus > 1.

(1) View Γ as a uniform lattice in SO(2, 1). Map SO(2, 1) → Sp [START_REF] Apanasov | Bending and stamping deformations of hyperbolic manifolds[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. This gives rise to a representation into Sp(2, 1) which can be deformed to a discrete Zariski-dense representation.

(2) View Γ as a uniform lattice in SU [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. Map SU(1, 1) → Sp(1, 1) → Sp [START_REF] Apanasov | Bending and stamping deformations of hyperbolic manifolds[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. This gives rise to a representation into Sp(2, 1) fixing a quaternionic line. Then there exists small deformations which do not stabilize any quaternionic line.

Whereas in the first case, explicit examples of deformations are provided by Thurston's bending construction, the existence of Zariski dense deformations in the second case follows from rather general principles. It would be interesting to visualize some of them.

1.4. Plan of the paper. Section 2 gives a cohomological criterion for non Zariski dense sugroups to remain non Zariski dense after deformation. The necessary cohomology vanishing is obtained in section 3. Theorem 1.1 is proved in section 4, Theorem 1.2 in section 5. The statements for nonuniform lattices are proved in section 6. Section 7 describes how lattices in Lie subgroups can sometimes be bent to become Zariski dense. The proof of Theorem 1.3 is completed in section 8. We end with a remark on non Zariski dense discrete subgroups in section 9.

A relative Weil theorem

Let Γ be a finitely generated group, and G be a Lie group with Lie algebra g. The character variety χ(Γ, G) is the quotient of the space Hom(Γ, G) of homomorphisms of Γ to G by the action of G by postcomposing homomorphisms with inner automorphisms. In [START_REF] Weil | On discrete subgroups of Lie groups[END_REF], A. Weil shows that a sufficient condition for a homomorphism ρ : Γ → G to define an isolated point in the character variety is that the first cohomology group H 1 (Γ, g ρ ) vanishes. In this section, we state a relative version of Weil's theorem.

Let H ⊂ G be an algebraic subgroup of G. Let χ(Γ, H, G) ⊂ χ(Γ, G) be the set of conjugacy classes of homomorphisms Γ → G which fall into conjugates of H. In other words, χ(Γ, H, G) is the set of G-orbits of elements of Hom(Γ, H) ⊂ Hom(Γ, G). If ρ ∈ Hom(Γ, H), the representation g ρ = ad • ρ on the Lie algebra g of G leaves the Lie algebra h of H invariant, and thus defines a quotient representation, which we shall denote by g ρ /h ρ . Proposition 2.1. Let H ⊂ G be real Lie groups, with Lie algebras h and g. Let Γ be a finitely generated group. Let ρ : Γ → H be a homomorphism. Assume that H 1 (Γ, g ρ /h ρ ) = 0. Then χ(Γ, H, G) is a neighborhood of the G-conjugacy class of ρ in χ(Γ, G). In other words, homomorphisms Γ → G which are sufficiently close to ρ can be conjugated into H.

Proof: Hom(Γ, G) is topologized as a subset of the space G Γ of arbitrary maps Γ → G. Let Φ : G Γ → G Γ×Γ be the map which to a map f : Γ → G associates Φ(f ) : Γ × Γ → G defined by Φ(f )(γ, γ ′ ) = f (γγ ′-1 )f (γ)f (γ ′ ). In other words, a map f ∈ G Γ is a homomorphism if and only if Φ(f ) = 1.
Consider the map Ψ :

G × H Γ → G Γ which sends g ∈ G and f : Γ → H to the map Ψ(g, f ) : Γ → G defined by Ψ(g, f )(γ) = g -1 f (γ)g.
We need prove that the image of Ψ contains a neighborhood of ρ in Φ -1 [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF].

The cohomological assumption gives information on the differentials of Φ and Ψ. The differential D ρ Φ is equal to -d 1 where d 1 denotes the coboundary C 1 (Γ, g ρ ) → C 2 (Γ, g ρ ). The differential of Ψ at g = e and f = ρ is given by

D (e,ρ) Ψ(v, η) = -d 0 v + η, where d 0 denotes the coboundary C 0 (Γ, g ρ ) → C 1 (Γ, g ρ ). Since, for all f ∈ H Γ , Φ(Ψ(g, f ))(γ, γ ′ ) = g -1 Φ(f )(γ, γ ′ )g, D ρ Φ • D (e,ρ) Ψ = 0. Conversely, if we assume that H 1 (Γ, g ρ /h ρ ) = 0, any θ ∈ C 1 (Γ, g ρ ) such that D ρ Φ(θ) takes values in the subalgebra h can be written θ = -d 0 v + η where v ∈ g and η ∈ C 1 (Γ, h ρ ), i.e. θ belongs to the image of D (e,ρ) Ψ.
Clearly, Hom(Γ, G) and Hom(Γ, H) are real analytic varieties. To analyze a neigborhood of ρ in them, it is sufficient to analyze real analytic of even formal curves t → ρ(t). In coordinates for G (in which H appears as a linear subspace), such a curve admits a Taylor expansion

ρ(t) = ∞ n=0 a j t j ,
where a 0 = ρ and for j ≥ 1, a j ∈ C 1 (Γ, g ρ ) is a 1-cochain. Then Φ(ρ(t)) = 1 for all t. Expanding this as a Taylor series gives

1 = Φ(ρ) + D ρ Φ(a 1 )t + (D ρ Φ(a 2 ) + D 2 ρ Φ(a 1 , a 1 ))t 2 + • • • , which implies that D ρ Φ(a 1 ) = 0, D ρ Φ(a 2 ) + D 2 ρ Φ(a 1 , a 1 ) = 0, . . . The first equation says that a 1 is a cocycle. So is a 1 mod h, therefore there exist v ∈ g and b 1 ∈ Z 1 (Γ, h ρ ) such that a 1 = -d 0 v + b 1 . Let t → g(t) be an analytic curve in G with Taylor expansion g(t) = 1 + vt + • • • . Then the Taylor expansion of ρ 1 (t) = g(t) -1 ρ(t)g(t) takes the form ρ 1 (t) = 1 + b 1 t + • • • .
In other words, up to conjugating, we arranged to bring the first term of the expansion of ρ(t) into h.

The second equation now reads

D ρ Φ(a 2 )+D 2 ρ Φ(b 1 , b 1 ) = 0. It implies that D ρ Φ(a 2 ) takes its values in h. Therefore there exist v ′ ∈ g and b 2 ∈ Z 1 (Γ, h ρ ) such that a 2 = -d 0 v ′ + b 2 . Conjugating ρ 1 (t) by an analytic curve in G with Taylor expansion 1 + v ′ t 2 + • • • kills v ′
and replaces a 2 with b 2 in the expansion of ρ 1 (t). Inductively, one can bring all terms of the expansion of ρ(t) into h. The resulting curve belongs to Hom(Γ, H). This shows that in a neighborhood of ρ, Hom(Γ, G) coincides with G -1 Hom(Γ, H)G. Passing to the quotient, χ(Γ, H, G) coincides with χ(Γ, G) in a neighborhood of the conjugacy class of ρ.

A cohomology vanishing result

3.1. Preliminaries. For basic information on quaternionic hyperbolic space and surveys, see [START_REF] Kim | Geometry on exotic hyperbolic spaces[END_REF][START_REF] Kim | Geometry of quaternionic hyperbolic manifolds[END_REF][START_REF] Pansu | Sous-groupes discrets des groupes de Lie : Rigidité, Arithméticité[END_REF].

We regard H n as a right module over H by right multiplication.

Viewing H = C ⊕ jC = C 2 , left multiplication by H gives C-linear endomorphisms of C 2 . So H * = GL 1 H ⊂ GL 2 C. Similarly (x 1 + iy 1 + j(z 1 + iw 1 ), • • • , x n + iy n + j(z n + iw n )) is identified with (x 1 + iy 1 , • • • , x n +iy n ; z 1 +iw 1 , • • • , z n +iw n ) so that H n = C 2n and GL n H ⊂ GL 2n C. A C-linear map φ : H n → H n is H-linear exactly when it commutes with j : φ(vj) = φ(v)j. Then it follows that if J = 0 -I n I n 0 , GL n H = {A ∈ GL 2n C : AJ = J Ā}.
Any element in GL n H can be written as α + jβ where α and β are 2n×2n complex matrices. If we write a vector in H n in the form X +jY where X, Y ∈ C n , the action of α + jβ on it is αX -βY + j(αY + βX).

So a matrix α + jβ in GL n H corresponds to a matrix in GL 2n C α -β β α .
In this paper, we fix a quaternionic Hermitian form of signature (n, 1) on

H n+1 as v, w = n i=1 vi w i -v n+1 w n+1 .
Then the Lie group Sp(n, 1, H) = Sp(n, 1), which is the set of matrices preserving this Hermitian form is

{A ∈ GL n+1 H : A * J ′ A = J ′ }, where J ′ = I n 0 0 -1 .
It is easy to see that its Lie algebra sp(n, 1) is the set of matrices of the form ImH Y X sp(n -1, [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF] ,

where Y + X * J n = 0, X, Y ∈ H n and J n = I n-1 0 0 -1 . So we get sp(n, 1) = ImH ⊕ H n ⊕ sp(n -1, 1).

Note that the adjoint action of the subgroup Sp(1) 0 0 Sp(n -1, 1) preserves this decomposition. The action on the H n component is the standard action, Sp(n -1, 1)H n Sp(1) -1 . Identifying H n+1 with C 2n+2 as above, it is easy to see that Sp(n, 1, H) is exactly equal to U(2n, 2) ∩ Sp(2n + 2, C), i.e. to the set of unitary matrices satisfying AJ = J Ā. Indeed, the symplectic form with respect to the standard basis of C 2n+2 is 0 A -A 0 and A = I n 0 0 -1 .

We will often complexify real Lie algebras. For any M ∈ gl(2n, C), one can write

M = 1 2 (M -J MJ) -i( 1 2 (iM + iJ M J)).
So it is easy to see that gl(n, H) = {A ∈ gl(2n, C) : AJ = J Ā} is complexified to gl(2n, C). It is well-known that u(2n, 2) ⊗ R C = gl(2n + 2, C) and sp(2n

+ 2, C) ⊗ R C = sp(2n + 2, C) × sp(2n + 2, C).
From these, we obtain that

sp(n, 1) ⊗ R C = sp(2n + 2, C).
We are particulary interested in

sp(1, 1) ⊗ R C = sp(4, C).
The quaternionic hyperbolic n-space H n H in the unit ball model is

{(x 1 , • • • , x n )|x i ∈ H, |x i | 2 < 1}.
It can be also described as a hyperboloid model

{X ∈ H n+1 : X, X = -1}/ ∼ where X ∼ Y iff X = Y Sp(1
  Sp(1) 0 0 a λ -a a -λ 2λ -a  
where a ≥ 1 is a positive real number, λ ∈ Sp(1) with Reλ = 1 a . These elements constitute the parabolic elements in the center {(t, 0)} of the Heisenberg group. A general parabolic element fixing a point (0, 1) at infinity and not stabilizing the quaternionic line {(0, H)}, is of the form

  * x -x * * * * * *   ,
with x = 0. These elements constitute the parabolic elements which do not belong to the center of the Heisenberg group.

Proof: The quaternionic line {(0, H)} in the hyperboloid model has coordinate (0, H, 1). To fix this line, it is not difficult to see that the matrix should have the form of

A =   * 0 0 * * 0 * * *   . Since its inverse J ′ A * J ′
also fixes the quaternionic line, it should have the form as in the claim. Now to prove the second claim, note that the matrix should satisfy the equation

  Sp(1) 0 0 0 a b 0 c d   (0, 1, 1) = λ(0, 1, 1) for λ ∈ Sp(1). Also it should satisfy A * J ′ A = J ′ .
From these, we obtain

a + b = λ c + d = λ |a| 2 -|c| 2 = |d| 2 -|b| 2 = 1 āb -cd = 0.
Then we get ā(λ-a)-c(λ-c) = 0. So (ā-c)λ = |a| 2 -|c| 2 = 1, and we get c = a-λ. Now we divide A by a since a is nonzero. Note that Aa -1 represents the same element in P Sp(2, 1). Then we can assume that a is a positive real number, conjugating A if necessary. The fact that Reλ = 1 a follows from the other two equations. So the result follows. In Heisenberg group {(t, z)|t ∈ ImH, z ∈ H}, the center {(t, 0)} is the (ideal) boundary of the quaternionic line {(0, H)}. So these parabolic elements stabilizing the quaternionic line belong to the center. See [START_REF] Kim | Marked length rigidity of rank one symmetric spaces and their products[END_REF].

To prove the last claim, we just note that A(0, 1, 1) = λ(0, 1, 1) should be satisfied. The parabolic elements not stabilizing the quaternionic line {(0, H)} should have nonzero x by the first case.

Raghunathan's theorem.

In this section we collect information concerning finite dimensional representations of so(5, C), which will be necessary for our main theorem. The basic theorem we will make use of is due to M.S. Raghunathan, [START_REF] Raghunathan | On the first cohomology of discrete subgroups of semi-simple Lie groups[END_REF]. Theorem 3.2. Let G be a connected semi-simple Lie group. Let Γ ⊂ G be a uniform irreducible lattice and ρ : (Γ ⊂ G) → Aut(E) a simple non-trivial linear representation. Then H 1 (Γ; E) = 0 except possibly when g = so(n + 1, 1) (resp. g = su(n, 1)) and the highest weight of ρ is a multiple of the highest weight of the standard representation of so(n + 1, 1) (resp. of the standard representation of su(n, 1) or of its contragredient representation).

In this theorem, Raghunathan used Matsushima-Murakami's result where L 2 -cohomology is used. We observe that as long as we use L 2cohomology, this theorem still holds for non-uniform lattices. This issue will be dealt with in section 6.

Standard representation of sp(4, C).

In the previous section, we used the symplectic form with respect to the standard basis of

C 4 Q =     0 1 0 0 -1 -1 0 0 1 0     .
Then the Lie algebra sp(4, C) consists of complex matrices A B C D such that

A t 1 0 0 -1 + 1 0 0 -1 D = 0, C t -1 0 0 1 + 1 0 0 -1 C = 0, B t 1 0 0 -1 + -1 0 0 1 B = 0.
Then an obvious choice of a Cartan subalgebra h is

    x 0 0 y 0 0 -x 0 0 -y     .
Let L 1 and L 2 ∈ h * be defined by L 1 (x, y) = x, L 2 (x, y) = y. Then the natural action of sp(4, C) on C 4 has the four standard basis vectors e 1 , e 2 , e 3 , e 4 as eigenvectors with weights L 1 , L 2 , -L 1 , -L 2 . The highest weight is L 1 .

3.4. Representation of so(5, C). We shall use the isomorphism of sp(4, C) to so(5, C). It arises from the following geometric construction.

Let V = C 4 and ω be the symplectic form defined as before. Then

∧ 2 V * ⊗ ∧ 2 V * → C α ⊗ β → α ∧ β ω ∧ ω , is a nondegenerate quadratic form P on ∧ 2 V * .
Here since both α ∧ β and ω ∧ ω are 4-forms, there is a constant c so that α ∧ β = cω ∧ ω, so the quotient should be understood as such a constant. Take the orthogonal complement W of Cω with respect to this quadratic form. Any matrix A acts on 2-forms as follows: Aα(v, w) = α(Av, Aw). Then Sp(4, C) leaves W invariant and acts orthogonally on it. This gives a map from Sp(4, C) to SO(5, C) = SO(W ), which turns out to be an isomorphism.

Next, we relate the choice of Cartan subalgebra for sp(4, C) made in the preceding paragraph to the standard choice for so(5, C).

We first compute the Lie algebra isomorphism derived from the group isomorphism.

Let

z 1 , z 2 , z 3 , z 4 be standard coordinates of C 4 so that dz 1 ∧ dz 3 + dz 4 ∧ dz 2 = ω. Let ω 6 = ω and ω 5 = dz 1 ∧ dz 2 + dz 3 ∧ dz 4 , ω 4 = dz 1 ∧ dz 4 + dz 2 ∧ dz 3 , ω 1 = i(dz 1 ∧ dz 4 -dz 2 ∧ dz 3 ), ω 2 = i(dz 1 ∧ dz 2 -dz 3 ∧ dz 4 ), ω 3 = i(dz 1 ∧ dz 3 -dz 4 ∧ dz 2 ). This is an orthonormal basis of ∧ 2 V * . Let A t ∈ Sp(4, C) so that A 0 = I and d dt | t=0 A t = X ∈ sp(4, C
). Then for one-forms α, β, one can figure out the action of X on two-forms to see that

X(α ⊗ β) = d dt | t=0 A t (α ⊗ β) = (Xα) ⊗ β + α ⊗ (Xβ). Then X(α ∧ β) = (Xα) ∧ β + α ∧ (Xβ).
To make computation easier, we choose a basis of W as

v 1 = ω 1 + iω 4 √ 2 , v 3 = ω 1 -iω 4 √ 2 , v 2 = ω 2 + iω 5 √ 2 , v 4 = ω 2 -iω 5 √ 2 , v 5 = ω 3 .
With respect to this basis, the symmetric bilinear form P has P (v 1 , v 3 ) = 1 = P (v 2 , v 4 ) = P (v 5 , v 5 ) and P (v i , v j ) = 0 for all other pairs. With respect to this P , one can easily see that a Cartan subalgebra of so(5, C) = so(W ; P ) can be chosen as the set of matrices of the form

      x 0 0 0 0 0 y 0 0 0 0 0 -x 0 0 0 0 0 -y 0 0 0 0 0 0       .
Let (x, y, z, w) denote a diagonal matrix in sp(4, C). Then one can easily compute that

(1, 0, -1, 0)v 1 = v 1 , (1, 0, -1, 0)v 3 = -v 3 , (1, 0, -1, 0)v 2 = v 2 , (1, 0, -1, 0)v 4 = -v 4 , (1, 0, -1, 0)v 5 = 0. Similarly (0, 1, 0, -1)v 1 = -v 1 , (0, 1, 0, -1)v 3 = v 3 , (0, 1, 0, -1)v 2 = v 2 , (0, 1, 0, -1)v 4 = -v 4 , (0, 1, 0, -1)v 5 = 0.
So the element,

    1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0     , in a Cartan subalgebra of sp(4, C)
corresponds to an element in a Cartan subalgebra of so(5, C),

h 1 =       1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0       . Similarly     0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1     corresponds to h 2 =       -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0       .
This representation under the isomorphism to sp(4, C) is different from the standard representation of so(5, C) on C 5 as we will see below. Proof: With respect to the symmetric bilinear form P as before, a Cartan subalgebra of so(5, C) is the set of diagonal matrices (x, y, -x, -y, 0) as noted above. Then the standard representation of so(5, C) on C 5 has eigenvectors, the standard basis e 1 , e 2 , e 3 , e 4 , e 5 , with eigenvalues

L 1 , L 2 , -L 1 , -L 2 , 0. This has the highest weight L 1 .
The standard representation of sp(4, C) on C 4 has the highest weight L 1 as we saw in the previous section. Note that the Cartan subalgebra of sp(4, C) is generated by the diagonal matrices (1, 0, -1, 0) and (0, 1, 0, -1) with dual basis L 1 and L 2 . Then under the isomorphism from sp(4, C) to so(Cω ⊥ ), these two diagonal matrices are mapped to diagonal matrices

h 1 = (1, 1, -1, -1, 0) and h 2 = (-1, 1, 1, -1, 0). Let L ′ 1 , L ′ 2 be the images of L 1 , L 2 under this isomorphism.
Then in terms of the standard dual basis L 1 , L 2 of the Cartan subalgebra of so(5, C),

L ′ 1 = L 1 + L 2 2 , L ′ 2 = L 2 -L 1 2 .
So the representation coming from the standard representation of sp(4, C) on C 4 has highest weight L 1 +L 2

2

. Actually this is the highest weight of the spin representation.

Corollary 3.4. Let Γ ⊂ Sp(1, 1) be a uniform lattice. Then H 1 (Γ, H 2 ) = 0 where H 2 is denotes the standard representation of Sp(1, 1) restricted to Γ.

Proof: View H 2 as C 4 with Sp(1, 1) acting on it. If we complexify the real Lie algebra sp(1, 1), we get sp(4, C). Since the standard representation of sp(4, C) on C 4 is different from the standard representation of so(5, C) on C 5 with highest weight L 1 , Theorem 3.2 (Theorem 1 of Raghunathan [START_REF] Raghunathan | On the first cohomology of discrete subgroups of semi-simple Lie groups[END_REF]) applies, and H 1 (Γ, H 2 ) = 0.

Proof of Theorem 1.1 (uniform case)

Let Γ ⊂ Sp(1, 1) be a uniform lattice. Denote by ρ the embedding

Γ → Sp(1, 1) → Sp(2, 1). Let G = Sp(2, 1), H = Sp(1) 0 0 Sp(1, 1) ⊂ G.
As was seen in section 3. ) as a subgroup which stabilizes a quaternionic line. Then every small deformation of Γ in Sp(n, 1) stabilizes a quaternionic line.

3-manifold case

In this section, we prove Theorem 1.2 for uniform 3-dimensional hyperbolic lattices. Let Γ ⊂ Spin(3, 1) 0 be a uniform lattice. According to Proposition 2.1, local deformations of the standard representation ρ 0 : Γ → Spin(3, 1) 0 → Spin(4, 1) 0 = Sp(1, 1) → Sp(2, 1) which do not stabilize a quaternionic line, are encoded in H 1 (Γ, H 2 ). We want to show that this first cohomology is zero. The complexified Lie algebra of SO(3, 1) is so(4, C). In the notations of section 3, the symmetric bilinear form P has a basis v 1 , v 2 , v 3 , v 4 so that P (v 1 , v 3 ) = P (v 2 , v 4 ) = 1 and P (v i , v j ) = 0 for all other pairs. The Cartan subalgebra of so(4, C) is the set of diagonal matrices (x, y, -x, -y). Then as in Lemma 3.3, the standard representation of so(4, C) on C 4 has a character which is not a multiple of the character of the representation coming from so(4, C) ⊂ sp(4, C). Then by Raghunathan's theorem 3.2, H 1 (Γ, H 2 ) = 0. Proposition 2.1 ensures that neighboring homomorphisms Γ → Sp(2, 1) stabilize a quaternionic line.

Non-uniform lattices

We used Raghunathan's theorem [START_REF] Raghunathan | On the first cohomology of discrete subgroups of semi-simple Lie groups[END_REF] to prove our main theorem when Γ is a uniform lattice. In this section we discuss how it generalizes, with restrictions, to nonuniform lattices.

The key point is whether Matsushima-Murakami's vanishing theorem that Raghunathan used still holds in non-uniform case. To apply Matsushima-Murakami's theorem, one has to use L 2 -cohomology.

Recall that under the subgroup

Sp(1) 0 0

Sp(1, 1) , the adjoint representation of Sp(2, 1) splits as a direct sum sp(2, 1) = sp(1) ⊕ H 2 ⊕ sp(1, 1). Let ρ denote the representation of Sp(1, 1) corresponding to the H 2 summand. Let M = H 4 R /Γ be a finite volume manifold. View Γ as a subgroup of Sp(1, 1), denote by ρ 0 the restriction of ρ to Γ. Let E be the associated flat bundle over M with fibre H 2 . It is well-known that E) is de Rham cohomology of smooth E-valued differential forms over M. We will denote this de Rham cohomology by H 1 (M, E).

H 1 (Γ, ρ 0 ) = H 1 dR (M, E) where H 1 dR (M,
In Matsushima-Murakami's proof, specific metrics on fibres of E, depending on base points, are used. More precisely, fix a maximal compact subgroup K of Sp(1, 1). Let sp(1, 1) = t ⊕ p be the corresponding Cartan decomposition. Fix a positive definite metric , F on H 2 so that ρ(K) is unitary and ρ(p) is hermitian symmetric. Then, for two elements v, w in the fibre over a point g ∈ G, one defines

v, w = ρ(g) -1 v, ρ(g) -1 w F .
Here is a concrete construction of such a metric on H 2 . As before, H 1,1 = H 2 is equipped with the signature (1, 1)-metric

Q = |q 1 | 2 -|q 2 | 2 .
Then for each negative H-line L in H 1,1 , there exists a positive definite H-Hermitian metric defined by -Q| L ⊕Q| L ⊥ where L ⊥ is the orthogonal complement of L with respect to Q.

A unit speed ray in H 4 R = H 1 H in terms of H 1,1 coordinates, can be written as l t = {q 1 = δ t q 2 } where δ t = e t -1 e t +1 , 0 ≤ t ≤ ∞. Note that here we normalize the metric so that its sectional curvature is -1. This can be easily computed considering a unit speed ray r(t) in a ball model emanating from the origin, and r(t) corresponds to the point (r(t), 1) in the hyperboloid model. Now we want to know how the metric varies along

l t as t → ∞. Let v = (v 1 , v 2 ) ∈ H 1,1 . It is easy to see that b t = ( 1 1 -δ 2 t , δ t 1 -δ 2 t
),

a t = ( δ t 1 -δ 2 t , 1
1 -δ 2 t )
are unit vectors on l ⊥ t , l t respectively. Then l t component of v is

( δ t v 2 + δ 2 t v 1 1 -δ 2 t , v 2 + δ t v 1 1 -δ 2 t ) and l ⊥ t component is ( δ t v 2 + v 1 1 -δ 2 t , δ 2 t v 2 + δ t v 1 1 -δ 2 t
).

Then it is easy to calculate the square of the length of v on l t , which is

1 + δ 2 t 1 -δ 2 t [|v 1 | 2 + |v 2 | 2 ] + 2 δ t 1 -δ 2 t (v 1 v 2 + v 2 v 1 ) = 2δ t 1 -δ 2 t |v 1 + v 2 | 2 + 1 -δ t 1 + δ t (|v 1 | 2 + |v 2 | 2 ).
In conclusion, the square of the length of v grows like e t |v| 2 along the ray l t in general. But for v 1 + v 2 = 0, it grows like e -t |v| 2 along the ray. This is the case when the deformation consists in parabolic elements fixing a point (0, 1) (in the ball model) and not stabilizing the quaternionic line {(0, H)}. See Lemma 3.1. These estimates will be used below.

Let M = M ≥ǫ ∪ M ≤η be the thick-thin decomposition of M so that η > ǫ and M ≤η is a standard cusp part of M. Assume for simplicity that the cuspidal part is connected. It is well-known that M ≤η is homeomorphic to T × R + with ds 2 = e -2r ds 2 T + dr 2 where T is a flat closed 3-manifold, r denotes distance from T × {0}, and

M ≥ǫ ∩ M ≤η is T × [0, 1].
Let π : T × R + → T be the projection on the first factor. Since

H k (T ) = H k (M ≤η ) by π * , we want to show that L 2 H k (M ≤η ) = H k (T ), to show that H k (M ≤η ) = L 2 H k (M ≤η ). Let α be a k-form on T . Then |π * α| ∼ e r 2
|α|e kr where r is the distance from the boundary of the thin part. Here e r 2 comes from the fibre metric and e kr comes from the base metric. Then

||π * α|| 2 L 2 = |α| 2 e 2kr+r e -3r ds T dr ≤ ||α|| 2 L 2 (T ) × C < ∞ if 2k + 1 < 3. So the pull-back form π * α is always a L 2 -form on M ≤η if α is a 0-form.
So we obtained Lemma 6.1. For a finite volume real 4-dimensional hyperbolic manifold M, H 0 (M ≤η , E) = L 2 H 0 (M ≤η , E).

Proof: For any α ∈ H * (T, E) = H * (M ≤η , E), its pull-back π * α is a L 2 -form on M ≤η for * = 0 as noted above. So any element in H 0 (M ≤η , E) has an L 2 -representative.

Unfortunately, we cannot conclude that H 1 (M, E) = L 2 H 1 (M, E). This hinders us from generalizing our theorem to non-uniform lattices. Our generalization involves a restriction on the representation. Proposition 6.2. Let M be a finite volume hyperbolic 3-manifold so that M = H 3 R /Γ. Then all small deformations of Γ ⊂ SO(3, 1) ⊂ Sp(1, 1) preserving parabolicity still stabilizes a quaternionic line. The same thing holds for a finite volume hyperbolic 4-manifold.

Proof: We give a proof only in dimension 3, since the 4-dimensional case can be obtained by the same method. Since M has finite volume, its boundary consists of tori T i . Let ρ 0 : π 1 (M) → Spin(3, 1) 0 ⊂ Sp(1, 1) ⊂ Sp(2, 1) be a natural representation.

If ρ t (π 1 (∂M)) is parabolic for all small t, by Lemma 3.1, it can contribute to the H 2 summand of sp(2, 1). But in this case, it can be represented by an L 2 form. The argument goes briefly as follows.

Let ρ t : π 1 (M) → Sp(2, 1) be an one-parameter family of deformations so that ρ t (π 1 (∂M)) is all parabolic. Let N be the ǫ-thick part of M. Then ∂N consists of tori and the universal cover of it in H 3 R are horospheres. Fix a component of ∂ N which is a horosphere H corresponding to a component T of ∂N. Conjugating ρ t by g t which depend smoothly on t if necessary, we may assume that ρ t (π 1 (T )) leaves invariant a common horosphere H ′ in H 2 H . Such a choice of g t is possible by the following argument. Let a be an element in π 1 (T ) such that all ρ t (a) are parabolic. The subset P of Sp(2, 1) consisting of parabolic elements is a smooth manifold at ρ 0 (a), and the map from P to ∂H 2 H associating to each element in P its unique fixed point is smooth in a neighborhood of ρ 0 (a).

We may assume that H ′ is based at (0, 1) (in the ball model). Then by Lemma 3.1, the contribution of this deformation to the H 2 summand is contained in the subset {(x, y)|x + y = 0} ⊂ H 2 . This will help us out.

Let ω be a differential form representing the infinitesimal deformation d dt ρ t on this cusp. Since ρ t (π 1 (T )) fixes (0, 1), ω takes its values in the subalgebra s ⊂ sp(2, 1) of Killing fields on H 2 H which vanish at (0, 1) and which are tangent to the horospheres centered at (0, 1). Therefore the norm of vectors of s decays along a geodesic pointing to (0, 1), at speed controlled by the maximal sectional curvature (in our case, which is the direction away from a quaternionic line, -1 4 ). In our situation, we are only concerned with the subspace

{(v 1 , v 2 )|v 1 + v 2 = 0} ⊂ H 2 .
So along the ray the squared norm decays like e -r |v| 2 asymptotically. Then integrating along a geodesic ray, we see that the 1-form ω defined on the cusp is in L 2 on the cusp. In more details, let the cusp be T ×[0, ∞) with coordinates (x, y, r), and the metric ds 2 = e -2r ds 2 T +dr 2 , then the volume form on this cusp is e -2r dS T dr. Note that we take a metric on H 3 R whose sectional curvature is -1. Then along [0, ∞), the orthonormal basis is {e r ∂ ∂x , e r ∂ ∂y , ∂ ∂r }. Then at (x, y, r), the norm of ω is

|ω(e r ∂ ∂x )| 2 + |ω(e r ∂ ∂y )| 2 since ω( ∂ ∂r ) = 0. So T ×[0,∞) ||ω|| 2 dV ol = ∞ 0 e -r e 2r e -2r T ||ω T || 2 dS T dr < ∞
where e -r comes from the norm decay on {(v 1 , v 2 )|v 1 + v 2 = 0}, e 2r comes from the decay of the metric on H 3 R along the ray (one should take an orthonormal basis {e r ∂ ∂x , e r ∂ ∂y , ∂ ∂r } along the ray). We do this for each cusp of M. Let ω i be a 1-form which is a L 2representative of the deformation d dt ρ t on the i-th cusp of M. Let α be a global 1-form representing the deformation d dt ρ t . Then ω i = α + dφ i where φ i is a function defined on the i-th cusp. Let φ be the union of φ i and ξ be a smooth function so that ξ = 1 on cusps and 0 outside cusps. Let

ω ′ = α + d(ξφ) = α + φdξ + ξdφ.
Then on each cusp,

ω ′ = α + dφ i = ω i . Thus ω ′ is in L 2 and [ω ′ ] = [α]
. Now again we can use Matsushima-Murakami's result for this case. See [START_REF] Koziarz | Harmonic maps and representations of non-uniform lattices of P U (m, 1)[END_REF][START_REF] Koziarz | Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type[END_REF] for a similar argument in complex hyperbolic space.

So we proved the theorem. We wonder whether the theorem holds without the assumption of preserving parabolicity.

Bending representations

Let G be an algebraic group. The Zariski closure of a subgroup H of G(R) is denoted by H.

Let X be a compact orientable hyperbolic n-manifold which splits into two submanifolds with totally geodesic boundary V and W , exchanged by an involution that fixes their common boundary. Such manifolds exist in all dimensions, [START_REF] Millson | On the first Betti number of a constant negatively curved manifold[END_REF]. Then Γ = π 1 (X) splits as an amalgamated sum Γ = A ⋆ C B where A = π 1 (V ), B = π 1 (W ) and C = π 1 (∂V ). Here, Ā = B = P O(n, 1) 0 and C = P O(n -1, 1) 0 . Now embed P O(n, 1) 0 into a larger group G. Let c belong to the centralizer Z G (C). Consider the subgroup Γ c = A ⋆ C cBc -1 . When c is chosen along a curve in Z G (C), one obtains a special case of W. Thurston's bending deformation, [START_REF] Thurston | The geometry and topology of 3-manifolds[END_REF] chapter 6. In this section, we analyze the Zariski closure of Γ c in case G = P Sp(m, 1) is the isometry group of m-dimensional quaternionic hyperbolic space, m ≥ n and P O(n, 1) 0 → P Sp(n, 1) → P Sp(m, 1) in the obvious manner.

7.1. The first bending step. We find it convenient to use a geometric language, and establish a dictionary between subgroups of G = P Sp(m, 1) and totally geodesic subspaces of

X = H m H . Lemma 7.1. The subgroup of G that leaves Y = H n R ⊂ X invariant is the normalizer of H = P O(n, 1) 0 in G. Proof: If aHa -1 = H, a maps the orbit Y of H to itself. Conversely, Y is the only orbit of H in X which is totally geodesic. If a ∈ G normalizes H, then a maps Y to itself.
Second, let us determine the space of available parameters for bending, i.e. elements which commute with C.

Proof: Clearly, Z G (C) = Z G (L). L stabilizes the totally geodesic subspace P = H n-1 R of the symmetric space X = H m H of G. If a ∈ G centralizes L,
then a normalizes it, thus it maps P to itself, by Lemma 7.1. Furthermore, the restriction of a to P belongs to the center of Isom(P ) = L, thus is trivial. In other words, a fixes each point of P . Conversely, isometries of X which fix every point of P centralize L and thus C. Indeed, L is generated by geodesic symmetries with respect to points of P , and these commute with isometries fixing P . To get the matrix expression of Z G (C), view X as a subset of quaternionic projective m-space. Then for every vector y ∈ R n , extended with zero entries to give a vector in R m+1 , there exists a quaternion q(y) such that a(y) = yq(y). This implies that a lifted as a matrix in Sp(m, 1) is block diagonal, a = qI n 0 0 D , with blocks of sizes n and m -n + 1 respectively, q ∈ Sp(1) and D ∈ Sp(m -n + 1). This product group maps to a subgroup of P Sp(m, 1) which is traditionnally denoted by Sp(m -n + 1)Sp( 1).

The dictionary continues with a correspondance between Zariski closures in simple groups and totally geodesic hulls in symmetric spaces.

Lemma 7.3. Let Y 1 , . . . , Y k be totally geodesic subspaces of a symmetric space X. Then Isom(Y j ) naturally embeds into G = Isom(X). Furthermore, the Zariski closure of j Isom(Y j ) equals Isom(Z) where Z is the smallest totally geodesic subspace of X containing j Y j .

Proof: For x ∈ X, let ι x denote the geodesic symmetry through x. Since X is symmetric, ι x is an isometry. Such involutions generate Isom(X). If Y ⊂ X is totally geodesic, then Y is invariant under all ι y , y ∈ Y . Therefore Y is again a symmetric space, with isometry group generated by the restrictions to Y of the ι y . In particular, the subgroup of G generated by the ι y , y ∈ Y , is isomorphic to Isom(Y ).

If γ is a geodesic joining points x ∈ Y i and y ∈ Y j , then ι x and ι y leave γ invariant. Their restrictions to γ generate an infinite dyadic group. The Zariski closure of this group contains all ι z where z ∈ γ. Therefore the Zariski closure of Isom(Y i )∪Isom(Y j ) contains ι z for all z belonging to the union of all geodesics intersecting both Y i and Y j . Since the totally geodesic closure Z is obtained by iterating this operation, one concludes that the Zariski closure of j Isom(Y j ) contains Isom(Z). Conversely, since Isom(Z) is an algebraic subgroup in G, it is contained in the Zariski closure.

Lemma 7.4. Let Y = H n R ⊂ H n H = X.
Let Z be a totally geodesic subspace of X such that Y Z X. Assume that Z contains a(Y ) where a ∈ G fixes pointwise a hyperplane P of Y but does not leave Y invariant. Then there is an isometry of X fixing Y pointwise and mapping Z to H n C . Proof: View the restriction of T X to Y as a vector bundle with connection ∇ on Y . Then T Z |Y is a parallel subbundle, therefore, for y ∈ Y , T y Z is invariant under the holonomy representation Hol(∇, y), which we now describe.

View Y as a sheet of the hyperboloid in R n+1 . Then a point y represents a unit vector, still denoted by y, in R n+1 . View X as a subset of quaternionic projective space. Then the point y also represents the quaternionic line Hy it generates. Such lines form the tautological quaternionic line bundle τ over X, a subbundle of the trivial bundle H n+1 equipped with the orthogonally projected connection. As a connected vector bundle, T X = Hom H (τ, τ ⊥ ). When restricted to Y , τ comes with the parallel section y. Therefore T X |Y = τ ⊥ = T Y ⊗ H. In other words, T X |Y splits as a direct sum of 4 parallel subbundles, each of which is isomorphic to T Y . It follows that Hol(∇, y) is the direct sum of four copies of the holonomy of the tangent connection, which is the full special orthogonal group SO(n). One of these copies is T y Y , the other are its images under an orthonormal basis (I, J, K) of imaginary quaternions acting on the right.

Let us show that Z contains a copy of H n C . Let a ∈ G fix a hyperplane P ⊂ Y pointwise. According to Lemma 7.2, F ix(P ) = Sp(1)Sp( 1), so a is given by two unit quaternions q and d. Pick an origin y ∈ P . Let u ∈ T y Y be a unit vector orthogonal to P . On T y X = T y Y ⊗ H, a acts by the identity on T y P and maps u to duq -1 . Since u is a real vector, a(u) = udq -1 ∈ T y Y ⊕ (T y Y )i where i = ℑm(dq -1 ). Up to conjugating by an element of the Sp(1) subgroup of G that fixes Y pointwise, one can assume that i is proportional to

I, i.e. T y Z contains uI. By assumption, uI / ∈ T y Y . By SO(n) invariance, T y Z contains T y Y ⊕ (T y Y )I = T y H n C , therefore Z contains Y ′ = H n C . Now T Z |Y ′ is a parallel subbundle of T X |Y ′ , thus T y Z is U(n)- invariant. Under U(n), T y X splits into only 2 summands. Since Z = X, T y Z = T y Y ′ , i.e. Z = Y ′ .
Along the way, we proved the following.

Lemma 7.5. Let Y ′ = H n C ⊂ H n H = X. Let Z be a totally geodesic subspace of X containing Y ′ . Then either Z = X or Z = Y ′ .
Corollary 7.6. After bending in P Sp(n, 1), a Zariski dense subgroup of P O(n, 1) 0 becomes Zariski dense in a conjugate of P U(n, 1).

Proof: Let Γ = A ⋆ C B be Zariski dense in P O(n, 1) 0 , with C Zariski dense in P O(n -1, 1) 0 . In other words, Γ leaves Y = H n R invariant, and C leaves P = H n-1 R invariant. Lemma 7.2 allows to select an a ∈ Z G (C) which does not map Y to itself. Lemma 7.4 shows that the smallest totally geodesic subspace of X = H n H containing Y and a(Y ) is congruent to H n C .
According to Lemma 7.3, this means that the bent subgroup A ⋆ C aBa -1 is Zariski dense in a conjugate of P U(n, 1). Therefore, to obtain a Zariski dense subgroup in P Sp(m, 1), m ≥ n, one must bend several times. 7.2. Further bending steps. We shall use compact hyperbolic manifolds which contain several disjoint separating totally geodesic hypersurfaces. Again, such manifolds exist in all dimension, see [START_REF] Millson | On the first Betti number of a constant negatively curved manifold[END_REF]. In low dimensions, a vast majority of known examples of compact hyperbolic manifolds have this property (they fall into infinitely many distinct commensurability classes, see [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]). Given such a manifold, bending can be performed several times in a row. The next lemmas show that at each step, the Zariski closure strictly increases.

Lemma 7.7. Let X ′ = H n H . Let Z be a totally geodesic subspace of X = H m H such that X ′ Z X.
Then Z is a quaternionic subspace. Furthermore, there exists an a ∈ G fixing X ′ pointwise which does not map Z into itself.

Proof: Otherwise, Z would be Sp(m -n)-invariant. In particular, for x ∈ X ′ , T x Z would be Sp(m -n)-invariant. Since Sp(m -n) acts irreducibly on (T x X ′ ) ⊥ , Z must be equal to X ′ or X, a contradiction. Z is a negatively curved symmetric space containing H n H , n ≥ 2, so it is a quaternionic subspace. Proof: According to Corollary 7.6, a first bending in P U(n, 1) provides us with a Zariski dense subgroup of P U(n, 1).

A second bending in P Sp(n, 1) gives a Zariski dense subgroup of P Sp(n, 1). Indeed, the fixator of H n-1 R is an Sp(1)Sp(1) which contains an element a which does not map H n C to itself. By Lemma 7.5, no proper totally geodesic subspace of H n H contains both H n C and a(H n C ). Lemma 7.3 implies that the bent subgroup is Zariski dense.

A third series of bendings gives a Zariski dense subgroup of P Sp(m, 1). Lemma 7.7 allows inductively to select a parameter a which strictly increases the dimension of the totally geodesic hull. After at most m -n more steps, the obtained subgroup is Zariski dense, thanks to Lemma 7.3. 7.3. Bending along laminations. Since we need to bend surfaces of genus as low as 2, which do not admit pairs of disjoint separating closed geodesics, we describe W. Thurston's general construction of bending along totally geodesic laminations, which does not require the leaves to be separating. We stick to the special case of totally real, totally geodesic 2-planes of The Lie algebra bundle B is a subbundle of the trivial bundle with fiber the Lie algebra sp(2, 1). Therefore, for every transversal curve c, the measure µ c can be pushed forward to yield an sp(2, 1)-valued measure on [a, b]. This measure integrates into a continuous map [a, b] → Sp(2, 1), see for example [START_REF] Estrade | Exponentielle stochastique et intégrale multiplicative discontinues[END_REF]. We denote the resulting element of Sp( 2 For instance, in the case of a discrete lamination, f is piecewise isometric and totally geodesic away from the support of λ. At each geodesic ℓ of the lamination, f bends, i.e. the totally geodesic pieces of the surface f (Y ) at either side of ℓ meet at a F ix(ℓ)-angle equal to exp(µ(ℓ)). The general case is best understood by considering limits of discrete measured laminations.

H 2 H . Let Y = H 2 R ⊂ H 2 H = X. If ℓ ⊂ Y is a geodesic, the subgroup F ix(ℓ) of Isom(X)
Let ρ : Γ → Sp(2, 1) be an isometric action of a group Γ which leaves Y and the measured lamination invariant. Then, for every piecewise transversal curve c, and γ ∈ Γ,

µ ρ(γ)(c) = ρ(γ)( µ c )ρ(γ) -1 . For γ ∈ Γ, let ρ λ (γ) = ( µ cγ )ρ(γ)
, where c γ is a piecewise transversal curve joining o to ρ(γ)o. Then ρ λ : Γ → Sp(2, 1) is a homomorphism which stabilizes f (Y ), and f is equivariant. Indeed, let c 1 (resp. c 2 ) be a piecewise transversal curve joining o to ρ(γ 1 )o (resp. to ρ(γ 2 )o).

Then c 1 ρ(γ 1 )(c 2 ) joins o to ρ(γ 1 γ 2 )o and

ρ λ (γ 1 γ 2 ) = ( µ c 1 ρ(γ 1 )(c 2 ) )ρ(γ 1 γ 2 ) = ( µ c 1 )( µ ρ(γ 1 )(c 2 ) )ρ(γ 1 γ 2 ) = ( µ c 1 )ρ(γ 1 )( µ c 2 )ρ(γ -1 1 )ρ(γ 1 γ 2 ) = ρ λ (γ 1 )ρ λ (γ 2 ).
If y ∈ Y and γ ∈ Γ, let c y (resp. c γ ) be a piecewise transversal curve joining o to y (resp. to ρ(γ)o). Then c γ ρ(γ)(c y ) joins o to ρ(γ)y, thus

f (ρ(γ)y) = ( µ cγρ(γ)(cy ) )ρ(γ)y = ( µ cγ )( µ ρ(γ)(cy ) )ρ(γ)y = ( µ cγ )ρ(γ)( µ cy )ρ(γ) -1 ρ(γ)y = ρ λ (γ)f (y).
Proposition 7.9. Let Σ be a closed hyperbolic surface with fundamental group Γ. Map Γ → SO(2, 1) → Sp(2, 1). There exist measured laminations λ on Σ which make the bent group ρ λ (Γ) Zariski dense in Sp(2, 1).

Proof: As a lamination, take the lifts to Y = Σ of two disjoint closed geodesics in Σ. A transversal measure in this case is simply the data of elements a j ∈ F ix(ℓ j ) for two lifts ℓ 1 , ℓ 2 . Note that the components of the complement of the two geodesics in Σ are not simply connected. In other words, each component of the complement of the support of the lifted lamination on Y is stabilized by a subgroup of Γ which is Zariski dense in SO(2, 1). It follows that the Zariski closure of ρ λ (Γ) contains SO(2, 1). It also contains the conjugates of SO(2, 1) by the two isometries a 1 and a 2 .

According to Lemma 7.3, the Zariski closure of ρ λ (Γ) contains the isometry group of the totally geodesic hull Z of Y ∪ a 1 (Y ) ∪ a 2 (Y ). As in the proof of Proposition 7.8, bending by a 1 gives a group which is Zariski dense in a conjugate of P U(2, 1), bending by a 1 and a 2 gives a group which is Zariski dense in P Sp(2, 1).

Flexibility of Fuchsian surface groups

In this section, we investigate homomorphisms of a surface group into Sp(2, 1) in a neighborhood of the embedding via SU(1, 1) and Sp(1, 1). We shall call them Fuchsian, to distinguish them from the bendable homomorphisms arising from the embedding via SO(2, 1). 8.1. Second order calculations. Let S be a compact Riemann surface with genus > 1 and ρ 0 : π 1 (S) = Γ ⊂ SU(1, 1) → Sp(1, 1) ⊂ Sp(2, 1) be a standard representation fixing a quaternionic line in H 2 H . Since H 1 (π 1 (S), H 2 ) = 0, Proposition 2.1 does not apply. We have to investigate which infinitesimal deformations represented by H 1 (π 1 (S), sp(2, 1)) are integrable.

The second order integrability condition for infinitesimal deformations at φ of representations of a group Γ in a Lie group G can be expressed in terms of the cup-product, a symmetric bilinear map

[•, •] : H 1 (Γ, g Adφ ) → H 2 (Γ, g Adφ ). For u ∈ Z 1 (Γ, g Adφ ), [u, u](α, β) = [u(α), Adφ(α)u(β)].
It is well-known, [START_REF] Nijenhuis | Deformations of homomorphisms of Lie groups and Lie algebras[END_REF], that for a representation φ from Γ to a reductive group G, if there exists a smooth path φ t in Hom(Γ, G) which is tangent to u ∈ Z 1 (Γ, g Adφ ), then [u, u] = 0. According to Theorem 3 in [START_REF] Goldman | Representations of fundamental groups of surfaces[END_REF], for surface groups, this necessary condition is also sufficient. Theorem 8.1. (W. Goldman). Let S be a closed surface, let G be a reductive group. Let φ : π 1 (S) → G be a representation such that the Zariski closure of φ(π 1 (S)) is also reductive. Then for any u ∈ Z 1 (π 1 (S), g Adφ ), [u, u] = 0 if and only if there exists an analytic path t → φ t in Hom(π 1 (S), G) which is tangent to u.

8.2.

Splitting of the cup-product map. The centralizer of SU(1, 1) in Sp(2, 1) is Sp(1) × U(1), where Sp(1) is the centralizer of Sp(1, 1) and U(1) ⊂ Sp(1, 1) is the centralizer of SU(1, 1) in Sp(1, 1). Then by Poincaré duality

H 2 (π 1 (S), sp(2, 1)) = H 0 (π 1 (S), sp(2, 1)) = sp(1) ⊕ u(1). Let u ∈ H 1 (π 1 (S), sp(2, 1)) split as u = u sp(1) + u sp(1,1) + u H 2 . Since sp(1, 1) and sp(1) commute, [u sp(1) , u sp(1,1) ] = 0. Since the subspace H 2 ⊂ sp(2, 1) is Sp(1)×Sp(1, 1)-invariant, [u sp(1) , u H 2 ] and [u sp(1,1) , u H 2 ] belong to H 2 (π 1 (S), H 2 ) = 0. Therefore [u, u] = [u sp(1) , u sp(1) ] + [u sp(1,1) , u sp(1,1) ] + [u H 2 , u H 2 ].
Since sp(1) and sp(1, 1) are subalgebras, [u sp [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF] , u sp(1) ] belongs to H 2 (π 1 (S), sp(1)) = sp(1), and [u sp [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF] , u sp(1,1) ] belongs to H 2 (π 1 (S), sp(1, 1)) = u(1). On the other hand,

[u H 2 , u H 2 ] has nontrivial components [u H 2 , u H 2 ] u(1) and [u H 2 , u H 2 ] sp(1)
on both H 2 (π 1 (S), sp(1, 1)) and H 2 (π 1 (S), sp(1)).

8.3. Homomorphisms to Sp(1). In the special case of the trivial representation to Sp(1), the cup-product map can be computed. Lemma 8.2. Let S be a closed surface. Let π 1 (S) act trivially on sp(1). The quadratic map H 1 (π 1 (S), sp(1)) → H 2 (π 1 (S), sp(1)), u → [u, u], is onto.

Proof: Here, H 1 (π 1 (S), sp(1)) ≃ H 1 (π 1 (S), R) ⊗ sp(1)), If a, b ∈ H 1 (π 1 (S), R) and q, q ′ ∈ sp(1), then [a ⊗ q, b ⊗ q ′ ] = a ⌣ b ⊗ [q, q ′ ].

For every q ′′ ∈ sp(1), there exist q, q ′ ∈ sp(1) such that [q, q ′ ] = q ′′ . Poincaré duality implies that there exist a, b ∈ H 1 (π 1 (S), R) such that a ⌣ b = 0. Therefore the cup-product map is onto. Let W denote the set of matrices of the form j z w -w t , where z, w and t ∈ C. Then W is a U(1, 1)-invariant complement of u(1, 1) in sp(1, 1). Given two elements X = j z w -w t and X ′ = j z ′ w ′ -w ′ t ′ in W , one computes π u(1) ([X, X ′ ]) = 2Im(zz ′ + tt ′ -2 ww ′ ). This is a symplectic structure on W (viewed as a real vector space). From Poincaré duality for local coefficient systems, it follows that the quadratic form π u(1) ([•, •]) on H 1 (π 1 (S), W ) is nondegenerate. In particular, it is onto. A fortiori, the quadratic form [•, •] on H 1 (π 1 (S), sp(1, 1)) is onto. 8.5. Flexibility of certain Fuchsian surface groups. A surface group in SU(n, 1) is Fuchsian if it stabilizes a complex line in complex hyperbolic space. Let us extend the terminology. Say a surface group in Sp(n, 1) is Fuchsian if it stabilizes a complex line in quaternionic hyperbolic space. Note that every complex line is contained in a unique quaternionic line.

It is well-known that Fuchsian groups in SU(2, 1) (or, more generally, SU(n, 1)) cannot be deformed to Zariski dense groups. We show that when SU(2, 1) is embedded in the larger group Sp(2, 1), this rigidity property fails. We make essential use of the main result of [START_REF] Goldman | Representations of fundamental groups of surfaces[END_REF].

Proposition 8.4. Let S be a compact Riemann surface with genus > 1 and ρ 0 : π 1 (S) = Γ ⊂ SU(1, 1) → Sp(1, 1) ⊂ Sp(2, 1) be a standard representation fixing a quaternionic line in H 2 H . Then there exist local deformations of ρ 0 which do not stabilize any quaternionic line.

Proof: Let u ∈ H 1 (π 1 (S), H 2 ) be nonzero. According to Lemmas 8.2 and 8.3, there exist v ∈ H 1 (π 1 (S), sp(1)) and w ∈ H 1 (π 1 (S), sp(1, 1)) such that [v, v] = -[u, u] sp(1) and [w, w] = -[u, u] u [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF] . Then x = u + v + w ∈ H 1 (π 1 (S), g) is nonzero and satisfies [x, x] = 0. According to Goldman's Theorem 8.1, there exists an analytic curve t → ρ t in Hom(π 1 (S), G), starting at ρ 0 , whose initial speed is a representative of the cohomology class x. Since x / ∈ H 1 (π 1 (S), sp(1) ⊕ sp(1, 1)), for t = 0 small, ρ t cannot be conjugated to the subgroup Sp(1, 1)Sp(1), i.e., does not stabilize any quaternionic line. Proof of Theorem 1.3. Proposition 8.4 is statement (2) of Theorem 1.3. Statement (1) of Theorem 1.3 is a consequence of the bending construction. For surfaces of sufficiently high genus, one can apply Proposition 7.8. In low genus, one needs bend along a geodesic lamination, see Proposition 7.9. 9. Discrete representations Proposition 9.1. Let Γ be a uniform lattice in Sp(1, 1). Let ρ : Γ → Sp(2, 1) be a discrete and faithful homomorphism. Then,

• either ρ is standard, i.e. it stabilizes a quaternionic line,

• or the image is Zariski dense.

Proof: Suppose ρ(Γ) is not Zariski dense. Then it cannot be contained in a parabolic subgroup of Sp(2, 1) since Γ is not solvable. So it must stabilizes a totally geodesic subspace of H 2 H , see [START_REF] Kim | Rigidity on symmetric spaces[END_REF]. If it stabilizes a quaternionic line, it is a standard representation, by Mostow rigidity. Suppose it stabilizes H 2 C . Then H 2 C /ρ(Γ) is a manifold. If it is not closed, the cohomological dimension of Γ cannot be 4, which contradicts Γ being a uniform lattice in Sp [START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF][START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. So H 2 C /ρ(Γ) is a closed manifold, which implies that H 2 C and H 4 R are quasi-isometric, which is impossible, again by a result of G.D. Mostow.

We suspect that there is no Zariski dense discrete faithful group ρ(Γ).

1 I.

 1 Kim gratefully acknowledges the partial support of KRF Grant (0409-20060066) and a warm support of IHES during his stay.

Lemma 3 . 3 .

 33 The highest weight of the standard representation of so(5, C) on C 5 is not a multiple of the highest weight of the representation coming from sp(4, C) on C 4 .

Lemma 7 . 2 .

 72 Let m ≥ n ≥ 2. Let L = P O(n -1, 1) 0 ⊂ P O(n, 1) 0 ⊂ P Sp(n, 1) ⊂ P Sp(m, 1) = G. Let C ⊂ L be a Zariski dense subgroup. Then the centralizer Z G (C) consists of isometries which fix P = H n-1 R pointwise. As a matrix group, Z G (C) = Sp(m -n + 1)Sp(1).

Proposition 7 . 8 .

 78 Let M be a compact hyperbolic n-manifold. Let m ≥ n. Assume that M contains N disjoint separating totally geodesic hypersurfaces. Let Γ = π 1 (M) ⊂ P O(n, 1) 0 → P Sp(m, 1). If N ≥ m -n + 2, then Γ can be continuously deformed to a Zariski dense subgroup of P Sp(m, 1).

  that fixes ℓ pointwise is conjugate to Sp(1)Sp[START_REF] Allcock | Infinitely many hyperbolic Coxeter groups through dimension 19[END_REF]. The Lie algebras of these subgroups form an ImH⊕ImH-bundle B over the space L of geodesics in Y . Pick once et for all an arbitrary Borel trivialization of this bundle. A lamination on Y is a closed subset of L consisting of pairwise non intersecting geodesics. A measured lamination on Y is the data of a lamination λ and a transverse ImH ⊕ ImH-valued measure. By a transverse measure, we mean the data, for each continuous curve c : [a, b] → Y which crosses all geodesics of λ in the same direction, of a finite Borel ImH ⊕ ImH-valued measure µ c on [a, b], with the following compatibility : if a curve c ′ : [a, b] → Y can be deformed to c by sliding along λ, then µ c ′ = µ c . A discrete collection of geodesics, with an ImH ⊕ ImH-valued Dirac mass at each geodesic, is a simple example of a measured lamination. Since only such laminations will ultimately be used, we shall not discuss non discrete measured laminations further.

, 1 )

 1 by µ c . If c = c 1 c 2 is obtained by traversing a first curve c 1 and then a second curve c 2 , then Chasles rule µ c 1 c 2 = ( µ c 1 )( µ c 2 ) holds, which allows to extend the definition to curves which are piecewise transversal. Define a map f : Y → X as follows. Pick an origin o ∈ Y . Given y ∈ Y , join o to y with a piecewise transversal curve c y and set f (y) = ( µ cy )y. One checks that f (y) does not depend on the choice of piecewise transversal curve.

8. 4 .

 4 Homomorphisms to Sp(1, 1). A similar statement applies to H 1 (π 1 (S), sp(1, 1)).

Lemma 8 . 3 .

 83 Let S be a closed hyperbolic surface. View π 1 (S) as a subgroup of SU(1, 1) ⊂ Sp(1, 1). The quadratic mapH 1 (π 1 (S), sp(1, 1)) → H 2 (π 1 (S), sp(1, 1)) = u(1), u → [u, u], is onto. Proof: sp(1,1) consists of quaternionic 2 × 2 matrices a b b d with a, d imaginary quaternions. The complex matrices in sp(1, 1) form the subalgebra u(1, 1) = su(1, 1) ⊕ u(1), where u(1) consists of complex imaginary multiples of the unit matrix. As a U(1, 1)-invariant projection sp(1, 1) → u(1) = R, we can use the linear form π u(1) a b b d = ℜe(i(a + d)).

  1, the adjoint representation of G restricted to H splits as a direct sum sp(2, 1) = sp(1) ⊕ H 2 ⊕ sp(1, 1), thus g/h = H 2 , restricted to Sp(1, 1), is the standard representation of Sp(1, 1). Corollary 3.4 asserts that H 1 (Γ, H 2 ) vanishes. Therefore H 1 (Γ, g ρ /h ρ ) = 0. According to Proposition 2.1, this implies that homomorphisms Γ → Sp(2, 1) which are close enough to ρ can be conjugated into H, i.e. leave a quaternionic line invariant.

	Since the subgroup of the form
		Sp(1) 0	0	
		0	I	0	
		0	0 Sp(1, 1)
	stabilizes a quaternionic line (0, 0, • • • , 0, H) in the ball model, we ob-tain

Corollary 4.1. Let Γ ⊂ Sp(1, 1) be a uniform lattice. Embed Γ into Sp(n, 1

Acknowledgement. We thank an anonymous referee for his valuable suggestions.