Marie Cottrell
email: cottrell@univ-paris1.fr

Patrick Letrmy

Missing values : processing with the Kohonen algorithm

Keywords: Data Analysis, Kohonen maps, Missing Values

We show how it is possible to use the Kohonen self-organizing algorithm to deal with data with missing values and estimate them. After a methodological reminder, we illustrate our purpose with three applications to real-world data. Nous montrons comment il est possible d'utiliser l'algorithme d'auto-organisation de Kohonen pour traiter des donnes avec valeurs manquantes et estimer ces dernires. Aprs un rappel mthodologique, nous illustrons notre propos partir de trois applications des donnes relles.

Introduction

The processing of data which contain missing values is a complicated and always awkward problem, when the data come from real-world contexts. In applications, we are very often in front of observations for which all the values are not available, and this can occur for many reasons : typing errors, fields left unanswered in surveys, etc.

Most of the statistical software (as SAS for example) simply suppresses incomplete observations. It has no practical consequence when the data are very numerous. But if the number of remaining data is too small, it can remove all significance to the results.

To avoid suppressing data in that way, it is possible to replace a missing value with the mean value of the corresponding variable, but this approximation can be very bad when the variable has a large variance.

So it is very worthwhile seeing that the Kohonen algorithm (as well as the Forgy algorithm) perfectly deals with data with missing values, without having to estimate them beforehand. We are particularly interested in the Kohonen algorithm for its visualization properties.

In Smal Ibbou's PHD thesis, one can find a chapter about this question, but it has not been published yet. The examples are run with the software written by Patrick Letrmy in IML-SAS and available on the SAMOS WEB page (http ://samos.univ-paris1.fr).

Adaptation of the Kohonen algorithm to data with missing values

We do not remind of the definition of the Kohonen algorithm here, see for example Kohonen [Kohonen, 1995], or [START_REF] Cottrell | Cartes autoorganises pour l'analyse exploratoire de donnes et la visualisation[END_REF].

Let us assume that the observations are real-valued p-dimensional vectors, that we intend to cluster into n classes.

When the input is an incomplete vector x, we first define the set M x of the numbers of the missing components. M x is a sub-set of {1, 2, . . . , p}.

If C = (C 1 , C 2 , ..., C n )
is the set of code-vectors at this stage, the winning code-vector C i0(x,C) related to x is computed as by setting

i 0 (x, C) = arg min i x -C i , where the distance x -C i 2 = k ∈Mx (x k -C i,k
) 2 is computed with the components present in vector x.

One can use incomplete data in two ways : a) If we want to use them during the construction of the code-vectors, at each stage, the update of the code-vectors (the winning one and its neighbors) only concerns the components present in the observation. Let us denote C t = (C t 1 , C t 2 , ..., C t n ) the code-vectors at time t and if a randomly chosen observation x t+1 is drawn, the code-vectors are updated by setting :

C t+1 i,k = C t i,k + ǫ(t)(x t+1 k -C t i,k )
for k / ∈ M x and j neighbor of i 0 (x t+1 , C t ). Otherwise,

C t+1 i,k = C t i,k .
The sequence ǫ(t) is [0,1]-valued with ǫ(0) ≃ 0.5 and converges to 0 as 1/t. After convergence, the classes are defined by the nearest neighbor method.

b) If the data are numerous enough to avoid using the incomplete vectors to build the map, one can content oneself with classifying them after the map is built, as supplementary data, by allocating them to the class with the code-vector which is the nearest for the distance restricted to non-missing components.

This method yields excellent results, provided a variable is not totally or almost totally missing, and also provided the variables are correlated enough, which is the case for most real data bases. Several examples can be encountered in Smal Ibbou's PHD thesis [Ibbou, 1998] and also in [START_REF] Gaubert | Segmented real estate markets and price mechanisms : the case of paris[END_REF].

Estimation of missing values, computation of membership probabilities

Whatever the method used to deal with missing values, one of the most interesting properties of the algorithm is that it allows an a posteriori estimation of these missing values.

Let us denote by C = (C 1 , C 2 , . . . , C n ) the code-vectors after building the Kohonen map. If M x is the set of missing component numbers for the observation x, and if x is classified in class i, for each index k in M x , one estimates x k by : xk = C i,k .

Because in the end of the learning the Kohonen algorithm uses no more neighbor (0 neighbor algorithm), we know that the code-vectors are asymptotically near the mean values of their classes. This estimation method therefore consists in estimating the missing values of a variable by the mean value of its class.

It is clear that this estimation is all the more precise as the classes built by the algorithm are homogeneous and well separated. Numerous simulations have shown as well for artificial data as for real ones, that when the variables are sufficiently correlated, the precision of these estimations is remarkable, [Ibbou, 1998].

It is also possible to use a probabilistic classification rule, by computing the membership probabilities for the supplementary observations (be they complete or incomplete), by putting :

P rob(x ∈ Class i) = exp(-x -C i 2 ) n k=1 exp(-x -C k 2 )
.

These probabilities also give confirmation of the quality of the organization in the Kohonen map, since significant probabilities have to correspond to neighboring classes.

Moreover, to estimate the missing values, one can compute the weighted mean value of the corresponding components. The weights are the membership probabilities. If x is an incomplete observation, and for each index k in M x , one estimates x k by :

xk = P rob(x ∈ Class i) C i,k .
These probabilities also provide confidence intervals, etc. In the following sections, we present three examples extracted from real data.

Socio-economic data

The first example is classical. The database contains seven ratios measured in 1996 on the macroeconomic situation of 182 countries. This data set was first used by F. Blayo and P. Demartines [START_REF] Blayo | Data analysis : How to compare kohonen neural networks to other techniques ?[END_REF] in the context of data analysis by SOMs.

The measured variables are : annual population growth (ANCRX), mortality rate (TXMORT), illiteracy rate (TXANAL), population proportion in high school (SCOL2), GDP per head (PNBH), unemployment rate (CHOMAG), inflation rate (INFLAT).

Among the set of 182 countries, only 115 have no missing values, 51 have only one missing value, while 16 have 2 or more than 2 missing values.

Therefore we use the 115 + 51 = 166 complete or almost complete countries to build the Kohonen map, and we then classify the 16 remaining countries. The data are centered and reduced as classically. We take a Kohonen map with 7 by 7 units, that is 49 classes. Figure 1 shows the contents of the classes. The 166 countries that were used for computing the code-vectors are in normal font, the 16 others in underlined italics. We can see that rich countries are in the top left hand corner, very poor ones are displayed in the top right hand corner. Ex-socialist countries are not very far from the richest, etc. As for the 16 countries which are classified after the learning as supplementary observations, we observe that the logic is respected. Monaco and Vatican are displayed with rich countries, and Guinea with very poor countries, etc.

From these computations, it is possible to calculate the membership probabilities of each supplementary observation of each of the 49 classes.

For example, the probabilities that Cuba belongs to class i are greater than 0.03 for classes i = (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (3, 3), (4, 3), (6, 3), (7, 3), the maximum (0.06) being reached for class (5, 2). We can notice (figure 1) that they are neighboring classes. From these probabilities, it is possible to estimate the distribution of the estimators of the missing values. For Cuba, the variables in question are GDP, Unemployment and Inflation.

From these results, it is possible (as it will be shown in the talk) to build super-classes by using an ascending hierarchical classification of the code-vectors and then to cross this classification with other exogenous classifications, etc.

Study of the property market in Ile-de-France

The second example is extracted from a study commissioned by the direction of Housing in the Regional Direction of Equipment in Ile-de-France (DHV/DREIF). This was achieved in 1993 by Paris 1 METIS and SAMOS laboratories, by [START_REF] Gaubert | Segmented real estate markets and price mechanisms : the case of paris[END_REF].

For 205 towns in Ile-de-France considered in 1988, we have property data (housing rents and prices, old and new, collective or individual, standard or luxurious, office rents and prices, old and new). Structurally, some of the data are missing, for example the office market can be nonexistent in some towns. This is a case where some data are structurally missing, and where the number of towns is dramatically reduced if one suppresses those which are incomplete : only 5 out of 205 would be kept ! So for the learning, we use 150 towns which have less than 12 missing values out of 15. After that, the 55 towns which have more than 12 missing values out of 15 are classified as supplementary observations.

Figure 2 displays the 205 towns (with and without missing values) classified on a 7 by 7 Kohonen map. Note that there are about 63% of missing values on the data set.

In this example which is practically impossible to deal with using classical software, we see that the Kohonen algorithm nevertheless allows to classify extremely sparse data, without introducing any rough error. The results are perfectly coherent, even though the data are seriously incomplete. The districts of Paris, Boulogne and Neuilly sur Seine are in the bottom left hand Of course, these good results can be explained by the fact that the 15 measured variables are well correlated and that the present values contain information about missing values. The examination of the correlation matrix (that SAS computes even in case of missing values) shows that 76 coefficients out of 105 are greater than 0.8, none of them being less than 0.65.

Structures of Government Spending from 1872 to 1971

The third example is a very classical one in data analysis, taken from the book "Que-sais-je ?" by Bouroche and Saporta , "L'analyse des donnes" [START_REF] Bouroche | L'analyse des Données[END_REF] On this projection, the years split up into three groups, which correspond to three clearly identified periods (before the First World War, between the two World Wars, after the Second World War). Only the year 1920, the first year when an expenditure item for Veterans appears, is set inside the first group, while it belongs to the second one. On the Kohonen map, the three super-classes (identical to the ones just defined) are identified by an ascending hierarchical classification of the code-vectors.

In this example, we have artificially suppressed randomly chosen values which were present in the original data, from 1 value out of 11 to 8 values out of 11, in order to study the clustering stability and compute the accuracy of the estimations that we get by taking the corresponding values of the code-vectors.

One can observe that the three super-classes remain perfectly stable as long as one does not suppress more than 3 values a year, that is 27% of the values.

Then we estimate the suppressed values in each case. The next table shows the evolution of the mean quadratic error according to the number of suppressed values. Tab.1 -Mean Quadratic Error according to the number of suppressed values

We notice that the error remains small as long as we do not suppress more than 3 values a year.

Conclusion

Through these three examples, we have thus shown how it is possible and desirable to use Kohonen maps when the available observations have missing values. Of course, the estimations and the classes that we get are all the more relevant since the variables are well correlated.

Example 2 shows that it can be the only possible method when the data are extremely sparse. Example 3 shows how this method allows to estimate the absent values with good accuracy. The completed data can then be dealt with using any classical treatment.

Fig. 1 -

 1 Fig.1 -The 182 countries (166 + 16) on a 7 by 7 map, 1500 iterations

Fig. 2 -

 2 Fig.2 -The 205 towns in Ile-de-France, in underlined italics the 55 towns which have more than 12 missing values out of 15 variables

  . The problem is to study the government spending, measured over 24 years between 1872 and 1971, by a 11-dimensional vector : Public Authorities (Pouvoirs publics), Agriculture (Agriculture), Trade and Industry (Commerce et industrie), Transports (Transports), Housing and Regional Development (Logement et amnagement du territoire), Education and Culture (Education et culture), Social Welfare (Action sociale), Veterans (Anciens combattants), Defense (Dfense), Debt (Dette), Miscellaneous (Divers). It is a very small example, with 24 observations of dimension 11, without any missing values.A Principal Component Analysis provides an excellent representation in two dimensions with 64% of explained variance. See figure3.

Fig. 3 -

 3 Fig.3-On the left, the projections on the first two principal axes ; on the right, the Kohonen map with 9 classes and 3 super-classes


Laurent Prévot
email: laurent.prevot@lpl-aix.fr

Jan Gorisch
email: gorisch@ids-mannheim.de

Sankar Mukherjee
email: sankar1535@gmail.com

Laurent Prvot

Annotation and Classification of French Feedback Communicative Functions

Feedback utterances are among the most frequent in dialogue. Feedback is also a crucial aspect of all linguistic theories that take social interaction involving language into account. However, determining communicative functions is a notoriously difficult task both for human interpreters and systems. It involves an interpretative process that integrates various sources of information. Existing work on communicative function classification comes from either dialogue act tagging where it is generally coarse grained concerning the feedback phenomena or it is token-based and does not address the variety of forms that feedback utterances can take. This paper introduces an annotation framework, the dataset and the related annotation campaign (involving 7 raters to annotate nearly 6000 utterances). We present its evaluation not merely in terms of inter-rater agreement but also in terms of usability of the resulting reference dataset both from a linguistic research perspective and from a more applicative viewpoint.

Introduction

Positive feedback tokens (yeah, yes, mhm ...) are the most frequent tokens in spontaneous speech. They play a crucial role in managing the common ground of a conversation. Several studies have attempted to provide a detailed quantitative analysis of these tokens in particular by looking at the form-function relationship [START_REF] Allwood | The MUMIN coding scheme for the annotation of feedback, turn management and sequencing phenomena[END_REF][START_REF] Petukhova | The independence of dimensions in multidimensional dialogue act annotation[END_REF][START_REF] Gravano | Affirmative cue words in task-oriented dialogue[END_REF][START_REF] Neiberg | Semisupervised methods for exploring the acoustics of simple productive feedback[END_REF]. About form, they looked at lexical choice, phonology and prosody. About communicative function, they considered in particular grounding, attitudes, turn-taking and dialogue structure management.

Despite the previous attempts to quantify that form-function relationship of feedback, we think that more work needs to be done on the conversational part of it. For example, [START_REF] Gravano | Affirmative cue words in task-oriented dialogue[END_REF] used automatic classification of positive cue words, however the underlying corpus consists of games, that are far off being "conversational" and therefore do not permit to draw any conclusions on how feedback is performed in conversational talk or talkin-interaction. What concerns the selection of the feedback units, i.e. utterances, more work that clarifies what consists of feedback is also needed, as an approach that purely extracts specific lexical forms ("okay", "yeah", etc.) is not sufficient in order to account for feedback in general. Also, the question of what features to extract (acoustic, prosodic, contextual, etc.) is far from being answered. The aim of this paper is to shed some more light on these issues by taking data from real conversations, annotating communicative functions, extracting various features and using them in experiments to classify the communicative functions.

The study reported in this paper takes place in a project [START_REF] Prévot | Cofee-toward a multidimensional analysis of conversational feedback, the case of french language[END_REF]) that aims to use, among other methodologies, quantitative clues to decipher the form-function relationship within feedback utterances. More precisely, we are interested in the creation of (large) datasets composed of feedback utterances annotated with communicative functions. From these datasets, we conduce quantitative (statistical) linguistics tests as well as machine learning classification experiments.

After presenting feedback phenomena and reviewing the relevant literature (Section 2), we introduce our dataset (Section 3), annotation framework and annotation campaign (Section 4). After discussing the evaluation of the campaign (Section 5), we turn to the feature extraction (Section 6) and our first classification experiments (Section 7).

Feedback utterances

Definition and illustration Concerning the definition of the term feedback utterance, we follow Bunt (1994, p.27): "Feedback is the phenomenon that a dialogue participant provides information about his processing of the partner's previous utterances. This includes information about perceptual processing (hearing, reading), about interpretation (direct or indirect), about evaluation (agreement, disbelief, surprise,...) and about dispatch (fulfillment of a request, carrying out a command, ...)."

As a working definition of our class feedback, we could have followed [START_REF] Gravano | Affirmative cue words in task-oriented dialogue[END_REF], who selected their tokens according to the individual word transcriptions. Alternatively, [START_REF] Neiberg | Semisupervised methods for exploring the acoustics of simple productive feedback[END_REF] performed an acoustic automatic detection of potential feedback turns, followed by a manual check and selection. But given our objective, we preferred to use perhaps more complex units that are closer to feedback utterances. We consider that feedback functions are expressed overwhelmingly through short utterances or fragments [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] or in the beginning of potentially longer contributions. We therefore automatically extracted candidate feedback utterances of these two kinds. Utterances are however already sophisticated objects that would require a specific segmentation campaign. We rely on a rougher unit: the Inter-Pausal Unit (IPU). IPUs are stretches of talk situated between silent pauses of a given duration, here 200 milliseconds. An example of an isolated feedback IPU is illustrated in Figure 1a. In addition to isolated items, we added sequences of feedback-related lexical items situated at the very beginning of an IPU (see section 3 for more details and Figure 1b for an example).

Related work

The study of feedback is generally associated with the study of back-channels [START_REF] Yngve | On getting a word in edgewise[END_REF], the utterances that are not produced on the main communication channel in a way not to interfere with the flow of the main speaker. In the seminal work by [START_REF] Schegloff | Discourse as an interactional achievement: Some use of uh-huh and other things that come between sentences[END_REF], back-channels have been divided into continuers and assessments. While continuers are employed to make a prior speaker continue with an ongoing activity (e.g. the telling of a story), assessments are employed to evaluate the prior speaker's utterance.

A formal model for feedback items was proposed by [START_REF] Allwood | On the semantics and pragmatics of linguistic feedback[END_REF]. It includes four dimensions for analysing feedback: (i) Type of reaction to preceding communicative act; (ii) Communicative status; (iii) Context sensitivity to preceding communicative act; (iv) Evocative function. The first dimension roughly corresponds to the functions on the grounding scale as introduced by [START_REF] Clark | Using Language[END_REF]: (contact / perception / understanding / attitudinal reaction). The second dimension corresponds to the way the feedback is provided (indicated / displayed / signalled). The third dimension, Context sensitivity, is divided into three aspects of the previous utterance: mood (statement / question / request / offer), polarity and information status of the preceding utterance in relation to the person who gives feedback. The fourth dimension, Evocative function, is much less developed but relates to what the feedback requires / evokes in the next step of the conversation.

Grounded in this previous work but more concerned with annotation constraints, especially in the context of multi-modal annotations, [START_REF] Allwood | The MUMIN coding scheme for the annotation of feedback, turn management and sequencing phenomena[END_REF] use a much simpler framework that is associated with the annotation of turn management and discourse sequencing. The feedback analysis is split into three dimensions: (i) basic (contact, perception, understanding); (ii) acceptance; (iii) emotion / attitudes that do not receive an exhaustive list of values but include happy, surprised, disgusted, certain, etc. [START_REF] Muller | An empirical study of acknowledgement structures[END_REF][START_REF] Muller | Grounding information in route explanation dialogues[END_REF] have focused on more contextual aspects of feedback: function of the feedback target and feedback scope.The work relies on a full annotation of communicative functions for an entire corpus. The annotations of feedback-related functions and of feedback scope are reported to be reliable. However, the dataset analysed is small. and the guide- lines are genre-specific (route instruction dialogues) while we intend here a generalisable approach.

More recent frameworks include work by [START_REF] Gravano | Affirmative cue words in task-oriented dialogue[END_REF] who propose a flat typology of affirmative cue word functions. This typology mixes grounding functions with discourse sequencing and other unrelated functions. It includes for example Agreement, Backchannel, discourse segment Cue-Beginning and Cue-Ending but also a function called Literal modifier. The reason for such a broad annotation is that every instance of an affirmative cue word is extracted following a completely formdriven simple rule. Such an approach allows to create high-performance classifiers for specific token types but hardly relates to what is known about feedback utterances in general. Their dataset is therefore much more homogeneous than ours in terms of lexical forms but more diverse in terms of position since we did not extract feedback related tokens occurring for example in a medial or final position of an IPU. A token-based approach forbids to give justice to complex feedback items such as reduplicated positive cue words, and obvious combinations such as ah ouais (=oh yeah), ok d'accord (=okeydoke). Their strategy is simply to annotate the first token and ignore the other. Our strategy is to capture potential compositional or constructional phenomena within feedback utterances. Moreover, even within a word-based approach, it is debatable to use space from a transcription to delineate the units of analysis. Some of these sequences could already be lexicalized within the actual spoken system. A final point concerns reduplicated words. It is often dif-ficult to determine whether an item is mh, mh mh or mh + mh. While treating IPUs does not completely resolve this issue, it is more precise than only annotating the first token.

The form-driven approach by Neiberg et al. ( 2013) also combines automatic data selection with lexical and acoustic cues. As for the function annotation, they identify five scalar attributes related to feedback: non-understanding -understanding, disagreement -agreement, uninterested -interested, expectation -surprise, uncertainty -certainty. This scalar approach is appealing because many of these values seem to have indeed a scalar nature. We adopt this two tier approach to characterize communicative functions. We first identify a BASE function and when this function is taken to hold some deeper evaluative content such as agreement or the expression of some attitude, a second level EVALUATION is informed. Moreover, our approach considers that a crucial aspect of feedback utterances is their contextual adequacy and dependence. To test this hypothesis, we included an annotation for the previous utterance in our annotation framework (more detail in section 4).

Dataset

All data used in this study come from corpora including conversational interactions (CID) and task oriented dialogues (Aix-MapTask). Both corpora include native French speaking participants.

CID Conversation Interaction Data (CID) are audio and video recordings of participants having a conversational interaction with the mere instruction of talking about strange things to kick-off the conversation [START_REF] Bertrand | Le CID-Corpus of interactional data-annotation et exploitation multimodale de parole conversationnelle[END_REF][START_REF] Blache | Multimodal annotation of conversational data[END_REF]. The corpus contains 8 hours of audio recordings 1 .

Aix-MapTask Remote The Aix-MapTask [START_REF] Bard | Aix Map-Task: A new French resource for prosodic and discourse studies[END_REF][START_REF] Gorisch | Aix Map Task corpus: The French multimodal corpus of task-oriented dialogue[END_REF] is a reproduction of the original HCRC MapTask protocol [START_REF] Anderson | The HCRC map task corpus[END_REF] in the French language. It involves 4 pairs of participants with 8 maps per pair and turning roles of giver and follower. The remote condition (MTR) contains audio recordings that sum up to 2h30 with an average of 6 min. 52 sec. per map 2 .

Data extraction Our objective is to obtain a dataset that covers as completely as possible feedback utterances. We exploited our rather precise transcriptions (aligned with the signal at the phone level with the tool SPPAS [START_REF] Bigi | SPPAS: a tool for the phonetic segmentation of speech[END_REF]) that include laughter, truncated words, filled pauses and other speech events. We started from the observation that the majority of feedback utterances are IPUs composed of only a few tokens. We first identified the small set of most frequent lexical items composing feedback utterances by building the lexical tokens distribution for IPUs made of three tokens or less. The 10 most frequent lexical forms are : ouais / yeah (2781), mh (2321), d'accord / agree-right (1082), laughter (920), oui / yes (888), euh / uh (669), ok (632), ah (433), voil / that's it-right (360). The next ones are et / and (360), non / no (319), tu / you (287), alors / then (151), bon / well (150) and then follow a series of other pronouns and determiners with frequency dropping quickly. We excluded tu, et and alors as we considered their presence in these short isolated IPUs were not related to feedback. We then selected all isolated utterances in which the remaining items were represented and treated now each IPU as an instance of our dataset. As mentioned in the introduction, we also extracted feedback related token sequences situated at the beginning of IPUs. This yielded us a total of more than 7000 candidate feedback utterances.

In terms of coverage, given our heuristics for selecting feedback utterances, we miss most of the 1 The CID corpus is available online for research: http: //www.sldr.org/sldr000027/en/.

2 The description of MTR is available online: http:// www.sldr.org/sldr000732.

short utterances that are uniquely made of repetitions or reformulations (not including feedback related tokens). Our recall of feedback utterances is therefore not perfect. However, our final goal is to combine lexical items with prosodic and acoustic features. Therefore, our heuristics focus on these tokens. About lexical items, our coverage is excellent. Although there are some extra items that are not in our list, such as (vachement (a slang version of 'a lot') or putain (a swear word that is used as a discourse marker in rather colloquial French), these items remain relatively rare and moreover, they tend to co-occur with the items of our list. Therefore, most of their instances are part of our dataset in the complex category.The plus sign in ouais+ and mh+ stands for sequences of 2 or more ouais or mh. The token complex corresponds to all other short utterances extracted that did not correspond to any item from the list, e.g. ah ouais d'accord, ah ben @ ouais,...). For more details on the dataset, see [START_REF] Prévot | A SIP of CoFee: A Sample of Interesting Productions of Conversational Feedback[END_REF].

Annotation of communicative functions

We ended up with 54733 cross-annotated candidate utterances from CID and MTR corpora. Although the initial annotation schema was fairly elaborate, not all the dimensions annotated yielded satisfactory inter-annotator agreement4 . In this paper we focus on two articulated dimensions: the BASE, which is the base function of the feedback utterance (contact, acknowledgment, evaluation-base, answer, elicit, other), and EVALUATION, which was informed when the evaluation-base value was selected as the BASE function (evaluations could be: approval, unexpected, amused, confirmation). The details for these two dimensions are provided in Table 1. We also asked annotators to rate what the function of the previous utterance of the interlocutor was (assertion, question, feedback, try, request, incomplete, uninterpretable). Although circular, this last annotation was gathered to tell us how useful this kind of contextual information was for our task.

To conduct the annotation campaign, seven undergraduate and master students were recruited. The campaign was realized on a duration of 2 months for most annotators. Annotating one feedback instance took on average 1 minute. We made sure that every instance received 3 concurrent annotations in order to be able to set-up a voting procedure for building the final dataset.

Evaluation

Inter-rater agreement

Concerning BASE value annotations, the average κ value for the best pair of raters for all the subdatasets with enough instances to compute this value was around 0.6 for both corpora: MTR (min: 0.45; max: 0.96) and CID (min: 0.4; max: 0.85). Multiκ yielded low values suggesting some raters were not following correctly the instructions (which was confirmed by closer data inspection). However, we should highlight that the task was not easy. There is a lot of ambiguity in these utterances and lexical items are only part of the story. For example, the most frequent token ouais could in principle be used to reach any of the communicative functions targeted. Even after close inspection by the team of experts, some cases are extremely hard to categorize. It is not even sure that the dialogue participants fully determined their meaning as several functions could be accommodated in a specific context.

While best pair's κ seems to be a very favorable evaluation measure, most of our samples received only 3 concurrent annotations. Moreover, aside a couple of exceptions, always the same two raters are excluded. As a result, what we call "best-pair kappa" is actually simply the removal of the annotation of the worse two raters from the dataset, which is a relatively standard practice. There could be a reason for these raters to behave differently from others. Because of timing issues, one annotator could not follow the training sessions with the others and had to catch up later. The other annotator did the training with the others but had to wait almost 2 months before performing the annotation.

Concerning EVALUATION values annotations, it is more complex to compute reliably an agreement to the sporadic nature of the annotation (evaluation values are only provided if the rater used this category in the BASE function). Since the set of raters that annotate a given sample varies, in most cases of MTR the number of instances annotated by a given set of raters is too small to compute reliably agreement. On the CID corpus, which has much larger samples, κ-measures of EVALUATION can be computed but exhibit huge variations with a low average of 0.3. This is indeed a difficult task since raters have to agree first on the BASE value and then on the value of the EVALUATION category. But, as we will see later, our voting procedure over cross-annotated datasets still yielded an interesting annotated dataset.

Quality of the reference dataset

In order to better understand the choice we have about data use and selection, we evaluated several datasets built according to different confidence thresholds.

For the base level, we started with the whole dataset and then built sub-datasets made of the same data but restricted to a certain threshold based on the number of raters that employ this category (threshold values: 1 3 , 1 2 , 2 3 , 3 4 , 1). More precisely, we computed a confidence score for each annotated instance. We then use these different datasets to perform two related tasks: classifying the functions of the whole dataset (using a None category for instances that did not reach the threshold) and classifying the functions within a dataset restricted to the instances that received an annotated category of a given threshold. In the case of the classification of eval, we first restricted the instances to the ones that received the evaluation value as value for the base category.

These datasets are ranging from noisy datasets (low threshold, full coverage) to cleaner ones but without full coverage. They correspond to two main objectives of an empirical study: (i) more linguistic / foundational studies would probably prefer to avoid some of the noise in order to establish more precise models to match their linguistic hypotheses, (ii) natural language engineering has no other choice than to work with the full dataset.

Composition of the dataset As for the BASE category distributions, the CID dataset is made of bit more than 40% of ack and eval, almost 15% of others and only 2% of answer (∼2%). The MTR dataset, has a similar amount of ack, about 20% of eval and answer, 10% of others and 5% of the elicit category (that was basically absent from CID).

As for the EVALUATION category, CID is mostly made of approbation (46%) and amused (38%), then confirmation (8%) and unexpected (6%) while MTR has over 60% confirmation, only 13% amused feedback and 17% approbation.

Feature extraction

For our experiments, we focused on speech data and our dimensions include properties of items themselves: lexical content (LEX), acoustics (ACO); and properties of their context: apparition, timing and position (POS). We also use three more dimensions: contextual information extracted automatically (CTX-AUT), supplied manually by our annotators (MAN)5 and meta-data (META). Some details about these features are provided here: LEX transcription string + presence vs. absence of frequent lexical markers (16 features before binarization) ACO pitch (min/max/stdev/height/span/steepness/ slope/NaN-ratio6 ), intensity (quartiles Q1, Q2, Q3), avg aperiodicity, formants (F1, F2, F3) and duration (16 features) POS speech environment in terms of speech/pause duration before/after the item for both the speaker and the interlocutor; including overlap information (10 features) CTX-AUT first/last tokens and bigrams of previous utterance and interlocutor previous utterance (18 features before binarization) MAN function of the interlocutor's previous utterance, a circular information providing a kind of topline (1 feature) META Corpus, Speaker, Session, Role (4 features) For the classification experiments, all textual and nominal features have been binarized. All numeric features have been attributed min max threshold values and then normalized within these thresholds.

7 Classification experiments

Classification of the Base function

Our first task was to classify the BASE function. The dataset we used most intensively was the one in which we retain only the base functions proposed by at least 2 3 of the annotators7 . This is computationally difficult because none of the levels involved is enough to perform this task. As we will see, only a combination of dimensions allows us to reach interesting classification scores.

We first compared the impact of the classifier choice on the dataset. We set-up a baseline consisting of the majority class for each frequent lexical item. For example, all single 'mh' are classified as ack because the majority of them are annotated with this function. Then, we took our full set of features (LEX+ACO+CTX-AUT) and ran many classification experiments with various estimators (Naive Bayes, Decision Tree, SVM and Ensemble classifiers -Ada Boost and Random Forest) that are part of the SCI-KIT LEARN Python library [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] and several parameter sets. The Random Forest method performed best. One explanation for this can be that Tree-based classifiers have no problem handling different categories of feature sets and are not confused by useless features. A nice consequence is that it becomes easy to trace which features contribute the most to the classification. This point is indeed crucial for us who intend to clarify the combination of the different linguistic domains involved. For this reason, and because all the experiments (varying various parameters) always ended up with an advantage for Random Forest, we used this classifier (with 50 estimators and minimum size of leaves of 10 instances) for the rest of the study in this paper.

We also checked the learning curve with this classifier and we have seen that it brings already interesting results with only one third of the dataset.

Our second task was to vary the sets of features used. We wanted however to refine this experiment by looking separately at each corpus. In figures 2a and 2b, the feature sets tested are the BASE-LINE described above, only LEXical, ACOustic or POSitional featues, the combination of the three (LPA), ALL automatically extracted features and ALL + MANually annotated previous utterance function. All experiments have been conducted with 10-fold cross-validation providing us the standard deviations allowing significance comparison as can be seen with the error bars in the figures (typically these deviations range between 1% to 2% for BASE and from 3% to 4% with some deviations going up to 10% for EVALUATION).

The results illustrated in Figure 2a, once we know what our features are good at, can be largely explained by the distribution of the categories across the corpora. There are therefore not many answer instances (∼2%) in this corpus, a category that is not well caught by our features yet. But LEX, POS and ACO are good to separate precisely ack, eval and other. The MTR dataset has much more answers, which explains the jump in f-measure if we add the manual annotation of the interlocutor's previous utterance (MAN). We simply did not manage to catch this contextual information with our features yet and this has a much stronger impact on MTR than on CID.

Classification of the evaluation function

We ran the same experiments for the EVALUATION category as presented in Figure 2b. The features used by the classifier are different. Within evaluation cases, POS becomes less informative while LEX and ACO retain their predictive power. Corpora differences explain the results. CID has much more AMUSED feedback that are well caught by lexical features. MTR has more confirmations that can be signalled by a specific lexical item (voilà) but that is also strongly dependent on which participant is considered to be competent about the current question under discussion.

Individual features contribution

A close inspection of some of the trees composing the Random Forest allows us to understand some of the rules used by the classifier across linguistic domains. Here are some of the most intuitive yet interesting rules:

• if acoustic values pitch span and F1 increase, attitudinal (EVAL) values are more likely than mere acknowledgment (ack) and this on various situations. • aperiodicity seems to have been used to catch amused values that would not be associated with a laughter in the transcription. • the presence of mh and laughter in the transcription is a very good predictor of ack (in the BASE task) and amused (in the EVAL task).

• with an increase of opb (silence duration of the interlocutor channel before the classified utterance), other than feedback is more likely.

Impact of the dataset's quality

We checked what happens when one varies the threshold used for proposing a label on the instances and the different results if one uses the whole dataset or only the instances that received a label at a given confidence score (lower score means more labelled data but more noise, higher score means less noise but also less labelled data). Unsurprisingly, the accuracy on the filtered dataset increases with the employed threshold. We note however that on the eval category that has a high score even with a low threshold, the accuracy gain is not fantastic.

About the non-filtered dataset, in the case of eval and at threshold > 3 4 , the classifier is focusing on the None category to reach a high score (since this category becomes dominant). As for base, we note that the changes in threshold have a complex effect on the accuracy. Accuracy is stable for the 1 3 to 1 2 shift (reliability on instances is better and coverage still very high), then decrease (with significant decrease of coverage). The shift from 3 4 to 1 shows a slight increase in accuracy (due to a better recognition of the None category).

Conclusions

In this paper, the focus was on communicative functions, as they are performed by conversational participants. For everybody who is not directly engaged in the conversation, it is difficult to distinctly categorise such behaviour. In fact, our classification results are getting close to the error rate of the naive raters themselves. On the one hand, we note that some basic important distinctions (in particular the ack vs. eval divide that can be related to [START_REF] Bavelas | Listeners as co-narrators[END_REF] generic vs. specific listener responses) can be fairly efficiently caught by automatic means. This is done thanks to the importance of lexical, positional and acoustic features in determining these differences. On the other hand, our system has to improve as soon as contextual information becomes more important like for identifying answer or confirmation.

This methodology is almost completely datadriven and can be therefore applied easily to other languages, given that the corresponding annotation campaign is realized. More precisely, the creation of our feature sets and extractions can be fully automated. The main processing step is the forcedalignment. Most of the lexical features can be derived by extracting token frequency from short IPUs (here 3 tokens or less). The real bottleneck is the annotation of communicative functions. But now that the general patterns are known, it becomes possible to design more efficient campaigns.

Figure 2 :

 2 Figure 2: Classification results: f-measure (y-axis) per feature set (x-axis).

Table 1 :

 1 Annotated categories of communicative functions and their paraphrases.

	Base Function	Paraphrase
	contact	I am still here listening.
	acknowledgment	I have heard / recorded what you said but nothing more.
	evaluation-base	I express something more than mere acknowledgement (approval,
		expression of an attitude,...).
	answer	I answer to your question / request.
	elicit	Please, provide some feedback.
	other	This item is not related to feedback.
	Evaluation
	approval	I approve vs. disapprove / agree vs. disagree with what you said.
	expectation	I expected vs. did not expect what you said.
	amusement	I am amused vs. annoyed by what you said.
	confirmation / doubt	I confirm what you said vs. I still doubt about what you said.

The difference from the original data points comes from missing annotation values and technical problems on some files.

Dimensions related to feedback scope and the structure of the interaction were not consistently annotated by our naive annotators and will not be discussed here further.

This corresponds to the annotation of the previous utterance of the interlocutor within this list of labels: assert, question, feedback, try (confirmation request), unintelligible, incomplete.

The ratio of unvoiced parts (NaN = Not a Number) and voiced parts of the F0 contour.

The majority of the instances have been cross-annotated by three annotators.

Acknowledgements

This work is supported by French ANR (ANR-12-JCJC-JSH2-006-01). The second author also benefits from a mobility from Erasmus Mundus Action 2 program MULTI of the European Union (GRANT 2010-5094-7). We would like to thank Roxane Bertrand for the help on the selection of feedback utterances, Brigitte Bigi for help with the automatic processing of the transcriptions and Emilien Gorene for help with recordings and annotation campaigns. Finally, we would like to thank all recruited students who performed the annotations.


Pierre Pansu

L p -cohomology of symmetric spaces *

Keywords: L p -cohomology, negative curvature, symmetric space, Besov space. Mathematics Subject Classification : 43A15, 43A80, 46E35, 53C20, 53C30, 58A14

This is a short survey of Riemannian geometric applications of L p -cohomology of thick spaces, p = 2.

1 What is L p -cohomology ?

Definition

Cohomology is the roughest invariant of topological spaces (much simpler than the fundamental group, for instance). To take a metric into account, one introduces decay conditions at infinity, this leads to L p -cohomology.

Definition 1 Let M be a Riemannian manifold. Let p > 1. Let L p Ω * (M ) denote the Banach space of L p -differential forms and Ω * ,p (M ) the space of L p -differential forms whose exterior differential is again L p , equipped with the norm ω Ω * ,p = ω p L p + dω p L p 1/p .

The complex (Ω * ,p (M ), d) has a cohomology H * ,p which is called the L pcohomology of M .

Reduced cohomology and torsion

When M is compact, the L p condition is no restriction, L p -cohomology coincides with usual de Rham cohomology. Therefore we shall concentrate on noncompact manifolds.

1

In general, H k,p = (Ω k,p ∩ Kerd)/dΩ k-1,p is not a Banach space, since dΩ k-1,p need not be closed. Therefore, one introduces a notation for its Hausdorff quotient.

Definition 2 R k,p = (Ω k,p ∩ Kerd)/dΩ k-1,p , T k,p = dΩ k-1,p /dΩ k-1,p . R k,p , the reduced cohomology, is a Banach space. The torsion T k,p does not have a norm. It is not a Hausdorff topological space.

Thick and thin ends

For Riemannian manifolds with thin ends (e.g. Riemann surfaces with finite area cusps), one expects L p -cohomology to coincide with the cohomology of some compactification, eventually with some small correction taking into account the specific metric behaviour of each end. This point of view has given rise to a huge litterature. We refer to S. Zucker's contribution to these proceedings.

The present notes concentrate on Riemannian manifolds with bounded geometry, i.e. with sectional curvature bounded on both sides and injectivity radius bounded from below. For short, we call them thick spaces. Examples of thick spaces are homogeneous spaces of Lie groups and universal coverings of compact manifolds.

We shall see that, for many thick spaces but not all, L p -cohomology is again expressible in terms of some compactification, but the analytic (and not only topological) properties of the compactification play a role.

1.4 Example: The real hyperbolic plane H 2 R Here H 0,p = 0 = H 2,p for all p. Let us begin with p = 2. Since the Laplacian on L 2 functions is bounded below, T 1,2 = 0. Therefore

H 1,2 = R 1,2 = {L 2 harmonic 1-forms} = {harmonic functions h on H 2 R with ∇h ∈ L 2 }/R .
Since the Dirichlet integral ∇h 2 in 2 dimensions is a conformal invariant, one can switch from the hyperbolic metric on the disk D to the euclidean metric on the disk. Therefore H 1,2 = {harmonic functions h on D with ∇h ∈ L 2 }/R = {Fourier series Σa n e inθ with a 0 = 0, Σ|n| |a n | 2 < +∞} which is the Sobolev space H 1/2 (R/2πZ).

More generally, for p > 1, T 1,p = 0 and H 1,p is equal to the Besov space B 1/p p,p (R/2πZ) mod constants.

In this example, hyperbolic plane is compactified into a disk, L p -cohomology identifies with a function space on the boundary circle.

Example: the real line

In that case, H 0,p = 0. R 1,p = 0 since every function in L p (R) can be approximated in L p with derivatives of compactly supported functions. Therefore H 1,p is only torsion. It is non zero and thus infinite dimensional. Indeed, the 1-form dt t (cut off near the origin) is in L p for all p > 1 but it is not the differential of a function in L p . This is an example where no compactification seems to help understanding L p -cohomology.

Functoriality

By definition, L p -cohomology is obviously invariant under biLipschitz diffeomorphisms. In the same way as cohomology is natural under continuous maps, and not only smooth maps, L p -cohomology is natural under a wider class of maps, called uniform maps, for thick spaces. Say a map f : X → Y between metric spaces is uniform if d(f (x), f (x ′ )) is bounded from above in terms of d(x, x ′ ) only. Proving this requires a modification of the definition, in order that it makes sense for rather general metric spaces. There are several possibilities, see [E2, F]. The following one is taken from [P4].

Definition 3 Let X be a metric space equipped with a measure. For each scale t > 0, consider the simplicial complex X t whose vertices are points of X and such that a subset ∆ ⊂ X of k + 1 points forms the vertices of a k-simplex if and only if its diameter is ≤ t. Simplicial cochains κ of X t possess an L p norm

κ p = Xו••×X |κ(x 0 , . . . , x k )| p dx 0 . . . dx k 1/p .
This allows to define L p -cohomology of X t .

Proposition 4 ( [P4]). Let X be a Riemannian manifold admitting a cocompact isometric group action. Assume that H j (X, R) = 0 for 1 ≤ j ≤ k. Then, for all t, H k,p (X t ) = H k,p (X). Furthermore, H k+1,p (X t ) = ker (H k+1,p

(X) → H k+1 (X, R)).
This means that L p -cohomology depends only on the Riemannian manifold viewed as a coarse metric space. The advantage of definition 3 is that it applies to discrete groups as well. One merely needs to select a left-invariant metric whose balls are finite sets.

Proposition 5 Let X, Y be measured metric spaces such that the volumes of balls are bounded above and below in terms of radius only. Then, for all k, for large enough t, there exists t ′ > 0 such that f induces a map f * :

H k,p (Y t ) → H k,p (X t ′ ). It follows that biuniform maps induce isomorphisms on L p -cohomology.
Note that any homomorphism between discrete groups is uniform. In other words, L p -cohomology is a functorial invariant of discrete groups which, in the case of uniform lattices in Lie groups, can probably be computed by analytic means. In this case, it is likely that the reduced part can be expressed in terms of some kind of boundary of the Lie group (compare A. Koranyi's contribution to these proceedings), but torsion cannot be excluded.

2 What is it good for : case p = 2 L 2 invariants constitute a rich theory by themselves, see [START_REF] Ück | L 2 -invariants : theory and applications to geometry and K-theory[END_REF]. We extract only a few illustrations of the meaning and use of L 2 -cohomology, in connection with geometry, in the thick case. We will not develop these points, and refer to the litterature.

Square integrable harmonic forms

If M is complete, Hodge theorem applies : R k,2 identifies with the space of square integrable harmonic k-forms. Therefore, it seems to be computable by analytic means, as the example of hyperbolic plane shows. In return, L 2 -cohomology of real, complex and quaternionic hyperbolic spaces, and especially its description in terms of differential forms on the boundary, has been used to compute K-theoretic invariants of the groups SO(n, 1), SU (n, 1) and Sp(n, 1), see [START_REF] Kasparov | Operator K-theory and its applications : elliptic operators, group representations, higher signatures, C * -extensions[END_REF][START_REF] Julg | Complexe de Rumin, suite spectrale de Forman et cohomologie L 2 des espaces symétriques de rang 1[END_REF][START_REF] Julg | Operator K-theory for the group SU (n, 1)[END_REF][START_REF] Julg | La conjecture de Baum-Connes pour le groupe Sp(n, 1)[END_REF].

Torsion in degree 1

Vanishing of T 1,2 (M ) is equivalent to the following isoperimetric inequality. There exists a constant C such that for all smooth, compactly supported functions u on M ,

u L 2 ≤ C du L 2 .
Under local bounded geometry assumptions, this is equivalent to the following isoperimetric inequality. There exists a constant C such that for all smooth, compact domains D ⊂ M ,

vol(D) ≤ C vol(∂D).
When M covers a compact manifold N with fundamental group Γ, isoperimetric inequality fails if and only if Γ is amenable, see [Gr]. Therefore amenable groups can be characterized by T 1,2 (M ) = 0, [Br].

Examples of amenable groups include Z (cf. the real line, above), solvable groups, groups of intermediate growth, see [GZ] for a state of the art.

Examples of non amenable groups include free groups, surface groups (cf. the hyperbolic plane above), nonelementary hyperbolic groups, lattices in Lie groups (except extensions of solvable Lie groups by compact Lie groups).

Group cohomology

Assume that M is contractible and covers a compact manifold with fundamental group Γ. Then H * ,2 (M ) identifies with the cohomology of the regular representation of Γ. Certain classes of groups are known to have vanising cohomology in degree 1. For instance, a finitely generated group Γ is a Kazhdan group if and only if H 1 (Γ, π) = 0 for all unitary representations π of Γ, see [DV]. For such a group, H 1,2 (M ) = 0.

Examples of Kazhdan groups include lattices in semi-simple Lie groups with no simple factors locally isomorphic to SO(n, 1) or SU (m, 1).

Examples of non Kazhdan groups include amenable groups, free groups, surface groups and lattices in SO(n, 1) or SU (m, 1).

1. Amenable groups have T 1,2 = 0 (see above) and R 1,2 = 0 (see 2.5 below).

2. Free groups and surface groups have T 1,2 = 0 and R 1,2 = 0.

3. Lattices in SO(n, 1) or SU (m, 1) have H 1,2 = 0 except for lattices in SO(2, 1) = SU (1, 1) which are surface groups.

Cohomology of towers of coverings

Assume again that M is contractible and covers a compact manifold N with fundamental group Γ. If nonzero, R k,2 is infinite dimensional. Nevertheless, one can define a kind of dimension by unit volume dim Γ (H k,2 ) known as the the k-th L 2 -Betti number b k,2 (N ) of N , [A]. This works for arbitrary manifolds, but in case Γ can be exhausted by a tower of finite index normal subgroups Γ j , the definition is easier : renormalized usual Betti numbers b k (Γ j \ M )/[Γ : Γ j ] converge to b k,2 (N ), [START_REF] Ück | Approximating L 2 invariants by their finite dimensional analogues[END_REF]. Thus R * ,2 reflects the behaviour of cohomology of large compact quotients. For expositions of L 2 -Betti numbers, see [START_REF] Ück | L 2 -invariants : theory and applications to geometry and K-theory[END_REF] and [P3]. For a connection between L 2 -Betti numbers and spaces with thin ends, see the series of papers by J. Cheeger and M. Gromov,[START_REF] Cheeger | On the characteristic numbers of complete manifolds of bounded curvature and finite volume[END_REF][START_REF] Cheeger | Bounds on the von Neumann dimension of Lsp 2-cohomoloy and the Gauss-Bonnet theorem for open manifolds[END_REF][START_REF] Cheeger | Chopping Riemannian manifolds Differential geometry[END_REF].

Euler-Poincaré characteristic of amenable groups

It has been observed early that abelian or nilpotent groups have vanishing Euler-Poincaré characteristic. It was not a trivial matter to extends this to solvable groups. In [START_REF] Cheeger | L 2 cohomology and group cohomology[END_REF], J. Cheeger and M. Gromov have extended this vanishing theorem to the wide class of amenable groups. Their proof relies on L 2 -cohomology. The additivity of dim Γ (it adds up exactly under direct sums) implies that the Euler-Poincaré characteristic of a group is equal to the alternating sum of its L 2 -Betti numbers. J. Cheeger and M. Gromov show that amenable groups have vanishing L 2 -Betti numbers. This follows easily from the isoperimetric characterization of amenable finitely generated groups. We cheat a bit : it is a delicate point to define L 2 -Betti numbers for arbitrary groups, not only those which admit a finite dimensional classifying space.

Discrete series

R * ,2 is a Hilbert space on which the isometry group of M acts unitarily.

In the case when M = G/K is a Riemannian symmetric space, R k,2 splits as the direct sum of irreducible representations which belong to the discrete series and have the same infinitesimal character as the trivial representation, [Bo]. This provides us with a concrete realisation of these discrete series representations. Fix a uniform lattice Γ ∈ G. Each a discrete series representation π has a Harish Chandra formal dimension w(π), which is proportional to its Γ-dimension,

w(π) = const.(G) dim Γ (π) vol(Γ \ G/K) .
L 2 -Betti numbers are thus proportional to sums of Harish Chandra formal dimensions.

Principal series

Principal series representations whose infinitesimal character is equal to that of the trivial representation contribute to the torsion part of L 2 -cohomology. Indeed, they are only weakly contained in the regular representation. This leads to the following theorem.

Theorem 1 (A. Borel, [Bo]). Let M = G/K be a Riemannian symmetric space.

1. If G and K have equal ranks, then

T * ,2 (M ) = 0 and R k,2 (M ) = 0 if and only if dim(M ) = 2k. 2. Otherwise, R * ,2 (M ) = 0 and T k,2 (M ) = 0 if and only if 2k ∈ (dim(M )- ℓ, dim(M ) + ℓ], where ℓ = rank C (G) -rank C (K).
This contribution to torsion can again be quantitatively measured. Back to the general case when M covers a compact manifold N with fundamental group Γ. All spectral projectors 1 (0,λ] (dd * ) have finite Γ-traces, therefore one can define Novikov-Shubin numbers

α k (N ) = lim sup λ→0 log tr Γ 1 (0,λ] (dd * ) log λ ,
which measure the polynomial decay of the spectral density function of dd * on k-forms, [NS].

Theorem 2 (M. Olbrich, [O], N. Lohoué, S. Mehdi [LM]).

Let M = G/K be a Riemannian symmetric space. Assume ℓ = rank C (G) -rank C (K) > 0. Then, for all k ∈ ( dim(M )-ℓ 2 , dim(M )+ℓ 2 ], the k-th Novikov-Shubin invariant is equal to α k = ℓ. Otherwise, the L 2 -spectrum of dd * |ker(d * ) ⊥ is bounded below.
3 What is it good for : case p = 2

The theory for p = 2 is much less advanced. What is missing is a generalization of dim Γ . Therefore no topological applications have been found yet.

Nevertheless, there exist significant geometric applications. They are less well known, therefore we shall develop them here.

Amenability again

The statements of paragraph 2.2 extend if we replace 2 by any p > 1. Therefore a finitely generated group is amenable if and only if T 1,p = 0 for some (resp. all) p > 1.

Euler-Poincaré characteristic of negatively curved manifolds

A longstanding conjecture, attributed to H. Hopf, claims that the Euler-Poincaré characteristic of a compact negatively curved 2m-manifold should be nonzero, and of the same sign as (-1) m . J. Dodziuk and I. Singer have proposed the following approach : prove that all L 2 -Betti numbers vanish, but the middle one, which is non zero, see [An,[START_REF] Yau | Problem section. Seminar on differential geometry[END_REF]. This program has been completed yet only for manifolds which admit an auxiliary Kähler metric, [G2], see section 4.

Hausdorff dimension at infinity

In degree 1, L p -cohomology is nondecreasing with p. Therefore, there is a critical p(M ) such that H 1,p (M ) = 0 for p < p(M ) and H 1,p (M ) = 0 for p > p(M ). It turns out that in the case of hyperbolic groups, this critical exponent can be interpreted as a kind of dimension of the ideal boundary, see [START_REF] Pansu | Dimension conforme et sphère à l'infini des variétés à courbure négative[END_REF][START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF][START_REF] Bourdon | Cohomologie ℓ p et espaces de Besov[END_REF]. Indeed, it is always less than or equal to the infimal Hausdorff dimension of metrics on the boundary compatible with the natural quasiconformal structure, with equality in many examples, including lattices in rank one simple Lie groups and Fuchsian hyperbolic buildings. However, the inequality may be strict, and this gives rise to examples of hyperbolic groups where the infimal Hausdorff dimension of metrics on the boundary is not achieved, see [BP] and section 5.

Curvature pinching

In higher degrees, torsion in L p -cohomology sometimes sharply captures negative sectional curvature pinching.

Let -1 ≤ δ < 0. Say a Riemannian manifold is δ-pinched if its sectional curvature lies between -a and δa for some a > 0. For example, real hyperbolic space

H n R is -1-pinched, complex hyperbolic space H m C , m ≥ 2, quaternionic hyperbolic space H m
H , m ≥ 2, and Cayley hyperbolic plane

H 2 O are -1 4 -pinched. For real hyperbolic spaces H n
R , torsion in L p -cohomology vanishes most of the time. In fact, in each degree, for at most one value of p, specificly, p = n-1 k-1 in degree k ≥ 2. This property extends to pinched Riemannian manifolds as follows.

Theorem 3 [P6]. If M n is simply connected and δ-pinched for some δ ∈ [-1, 0), then

p < 1 + n -k k -1 √ -δ ⇒ T k,p (M ) = 0.
This is sharp. Indeed, for every n ≥ 3, 2 ≤ k ≤ n -1 and δ ∈ [-1, 0), there exists ǫ > 0 and a δ-pinched homogeneous Riemannian manifold whose torsion does not vanish for p ∈ (1

+ n-k k-1 √ -δ -ǫ, 1 + n-k k-1 √ -δ
). However, this comparison theorem is not sharp for negatively curved symmetric spaces. For instance, for complex hyperbolic space H m C , for k = 2, the pinching comparison theorem predicts that torsion vanishes for p < m, whereas it turns out that torsion still does not vanish for m ≤ p < 2m, see section 6.

In other words, our apparently rough invariant, torsion in L p -cohomology, not only detects subtle matters like sectional curvature pinching, but also distinguishes between homogeneous spaces which satisfy the same curvature bounds.

Euler characteristic of negatively curved manifolds 4.1 Hopf's conjecture

If M is a compact Riemannian manifold with constant sectional curvature -1, with even dimension n = 2m, then its Euler-Poincaré characteristic is

χ(M ) = (-1) m 2 vol(S 2m ) vol(M ).
Indeed, the Chern-Weil integrand P χ (R) for the Euler class is a homogeneous polynomial of degree m in the curvature tensor R. The curvature tensors of the round sphere R 1 and of real hyperbolic space R -1 differ by a sign, R -1 = -R 1 . Therefore P χ (R -1 ) = (-1) m P χ (R 1 ). Integrating this relates Euler charateristics and volumes of constant curvature spaces.

A similar calculation applies to other rank one locally symmetric manifolds M (the sphere is replaced by projective spaces over the complex, quaternion and octonion numbers). In all cases, (-1) m χ(M ) > 0.

If m = 1, the Gauss-Bonnet formula χ(M ) = M R shows that χ(M ) < 0 as soon as R < 0. In higher dimension, it is not so clear which negativity assumption on curvature should imply a sign for the Euler characteristic. A longstanding conjecture, attributed to H. Hopf, claims that the Euler-Poincaré characteristic of a compact Riemannian 2m-manifold should be nonzero, and of the same sign as (-1) m , if its sectional curvature is negative.

The Dodziuk-Singer conjecture

It follows readily from the definition of L 2 -Betti numbers b i,2 (M ) that they can be used to compute Euler characteristic,

χ(M ) = n i=0 (-1) i b i,2 (M ).
This has led J. Dodziuk and I. Singer to conjecture the following. Let M be an n-dimensional compact Riemannian manifold with negative sectional curvature. Let M denote its universal covering. Then the reduced L 2cohomology R i,2 ( M ) vanishes if i = n/2, and does not vanish if i = n/2.

Note that the fact that M admits a discrete cocompact isometric group action is essential, as shown by M. Anderson, [An]. Indeed, there exist simply connected complete negatively curved manifolds which admit nonzero L 2 -harmonic forms in several degrees simultaneously. According to [P5] Théorème G, there even exist Riemannian homogeneous spaces which admit nonzero L 2 -harmonic forms in all degrees 3, 4, . . . , n -3 simultaneously.

Gromov's result

Theorem 4 (M. Gromov, [G2]). Let M be a compact manifold with dimension n = 2m. Assume that M admits a Kähler metric. Assume that the fundamental group of M is Gromov-hyperbolic. Then (-1) m χ(M ) > 0.

In the rest of this section, we shall give a proof of this theorem.

Role of the Kähler condition

A symplectic structure on a manifold is a differential 2-form ω such that 1. ω is nondegenerate, i.e. its matrix in an arbitrary basis of the tangent space is nonsingular.

2. dω = 0.

Say a Riemannian metric and a symplectic form are compatible if at each point, there exists an orthonormal basis of the tangent space in which the matrix of ω takes the block form 0 I -I 0 .

Definition 6 A Kähler metric on a manifold is the data of a Riemannian metric and parallel compatible symplectic structure.

The metric and symplectic structure together determine an integrable complex structure on M . Clearly, complex manifolds are rare among manifolds. Therefore, admitting a Kähler metric is very restrictive. Nevertheless, interesting examples abound, since complex projective space admits a Kähler metric, which restricts to a Kähler metric on every smooth complex submanifold.

The next proposition collects the two properties of Kähler manifolds that we shall need. See for instance [W], [Ba].

Proposition 7 Let (M, g, ω) be a Kähler manifold of dimension n = 2m. Denote by L : α → ω ∧ α the wedging with ω operator. Then 1. For all i < m, L m-i is bijection of i-forms to 2m -i-forms. In particular, L is injective in degrees < m.

L commutes with the Laplacian on differential forms.

This implies that L induces an injection on de Rham cohomology spaces

H i (M, R) → H i+2 (M, R) for i < m.
Here is an L 2 variant of this statement.

Corollary 8 Let (M, g, ω) be a complete Kähler manifold of dimension n = 2m. Then wedging with ω induces an injection on reduced L 2 -cohomology spaces R i,2 (M ) → R i+2,2 (M ) for i < m.

The role of negative curvature

A simply connected nonpositively curved manifold is contractible. In fact, it possesses canonical deformation retractions : fix an origin o, and move an arbitrary point along the unique geodesic segment joining it to o. Therefore, any cycle c bounds a canonical chain, the cone on c with vertex o. Here is a variant of this construction. In a simply connected nonpositively curved manifold, from any point, one can draw a geodesic ray asymptotic to a given geodesic ray γ. This allows to construct the cone cone γ (c) on a chain c, as the union of all rays emanating from points of c, asymptotic to γ.

γ c cone (c) γ
We need a bound on the volume of this chain. Negative sectional curvature means that asymptotic geodesics converge exponentially. I.e., if t → γ 1 (t), γ 2 (t) are geodesics such that the distance d(γ 1 (t), γ 2 (t)) tends to zero, then

d(γ 1 (t), γ 2 (t)) ≤ e -ηt d(γ 1 (0), γ 2 (0)),
where -η 2 is the upper bound of sectional curvature.

Thus, when sectional curvature is negative, the cone cone γ (c) is exponentially thin, therefore its volume is proportional to the volume of the chain,

vol(cone γ (c)) ≤ 1 η vol(c), provided dim(c) ≥ 1.
Dually, the cone construction applies to differential forms. Since the norm on forms dual to the volume of chains is the L ∞ norm, one gets Proposition 9 (M. Gromov). Let M be a complete simply connected Riemannian manifold with sectional curvature less than -η 2 < 0. Let i ≥ 2.

1. Every i -1-cycle c spans a i-chain z with vol(z) ≤ 1 η vol(c).

Every closed bounded differential

i-form α on M is the differential of a bounded differential i -1-form β with β L ∞ ≤ 1 η α L∞ .
Corollary 10 Assume M covers both a compact 2m-dimensional Kähler manifold and a compact negatively curved Riemannian manifold. Then 1. R i,2 ( M ) = 0 for all i = m.

2. T i,2 ( M ) = 0 for all i. Let us show that the spectrum of the Laplacian on i-forms is bounded below for all i = m (resp. orthogonally to harmonic forms in degree i = m). Let α be a i-form with i > m. Let γ be the the 2m -i-form such that α = L i-m γ. Pointwise, |γ| ≤ const.|α|. Since L i-m commutes with the Laplacian, and its inverse is bounded, ∆γ, γ ≤ const. ∆α, α .

Then α = d(β ∧ L i-m-1 γ) -β ∧ L i-m-1 dγ, α 2 = α, d(β ∧ L i-m-1 γ) -α, β ∧ L i-m-1 dγ ≤ const.( d * α γ + α dγ ) ≤ const. α ∆α, α .
This shows that the spectrum of the

L 2 Ω m ( M ) = ker(∆) ⊕ im(d) ⊕ im(d * )
(where closures are unneeded) gives the Laplace spectrum bound orthogonally to harmonic forms in degree m. This shows that d has a closed image in degree m as well. Therefore torsion vanishes. q.e.d.

Zero in the spectrum

To complete the proof of Gromov's theorem, one needs show that reduced L 2 -cohomology does not vanish in middle dimension. In view of Corollary 10, if not, L 2 -cohomology would vanish in all degrees. In other words, 0 would not belong to the spectrum of the Laplacian on forms in any degree. It turns out that this rarely happens for contractible spaces, as observed by J. Lott, [Lt]. In fact, it follows from M. Gromov and B. Lawson's relative index theorem, [GL], that 0 belongs to the spectrum for complete simply connected nonpositively curved manifolds with cover compact manifolds, see [Lt].

We cheated a bit, since Gromov's theorem has a weaker assumption : no negative curvature, only a Gromov-hyperbolic fundamental group. The proof requires two changes. First, the coning proposition has to be extended to hyperbolic metric spaces. Second, the relative index theorem has to be replaced by an adhoc avatar of the index theorem, which applies to manifolds with a cocompact action of a disconnected, nondiscrete Lie group, see [G2], [P3].

For variants and generalizations of Gromov's argument, see chapter 8 of [Ba] and references therein.

Conclusion

The main ingredients in Gromov's argument are

1. The cup-product H 2,∞ ⊗ H i,2 → H i+2,2 .
2. The Kähler package, which gives full power to cup-product with the Kähler class.

3. A vanishing theorem for L ∞ -cohomology, which follows from the contracting character of canonical deformation retractions.

Dimension of the ideal boundary

We have seen (Proposition 9) that, for simply connected negatively curved manifolds, L ∞ -cohomology vanishes in all degrees i ≥ 2. Note that H 1,∞ is never zero for an unbounded space. In this section, we shall exploit the fact that, for simply connected negatively curved manifolds, H 1,p is nonzero for p large enough.

Non vanishing of cohomology

If M is simply connected, L p -cohomology in degree 1 is isomorphic to the space of functions u on M whose gradient is in L p , modulo additive constants. Therefore, to produce non trivial L p -cohomology classes, one merely needs functions whose gradient is in L p , and which do not tend to a constant at infinity. When M is negatively curved, one can proceed as follows. In polar coordinates, the metric takes the form dr 2 + g r , where g r is a family of Riemannian metrics on the sphere. If sectional curvature satisfies 1

≤ K ≤ δ < 0, then ( sinh(r √ -δ) √ -δ ) 2 g 0 ≤ g r ≤ sinh(r) 2 g 0 ,
where g 0 denotes the round metric on the unit sphere. In follows that the volume of the sphere of radius r is at most const. sinh(r) n-1 . Let v be a smooth function on the sphere. Extend v to a radial function u on M (i.e. v does not depend on r). Then

|∇u| ≤ ( sinh(r √ -δ) √ -δ ) -1 .
Multiply u by a function of r so as to make it vanish near the origin. Then

M |∇u| p ≤ const. +∞ 1 ( sinh(r √ -δ) √ -δ ) -p sinh(r) n-1 dr,
i.e. ∇u ∈ L p as soon as p > n-1 √ -δ . On the other hand, if v is not constant, the radial extension is not in L p , even up to an additive constant. Thus the cohomology class of u is nonzero. We conclude that

p > n -1 √ -δ ⇒ H 1,p (M ) = 0.
More generally, Bourdon and Pajot show that, for every nonelementary hyperbolic group, R 1,p = 0 for p large enough.

Boundary value

Conversely, following [S], one shows that every function f on M whose gradient is in L p behaves asymptotically like u, i.e. has a boundary value v f , i.e. a limit along almost every ray. Indeed, because of exponential growth of volume, the radial derivative ∂f ∂r is L p with an exponential weight, thus is L 1 (Hölder inequality). Clearly, if f is L p , v f vanishes almost everywhere. Furthermore, if u f is the radial extension of v f , then f -u f is in L p (Hardy inequality, see [GKS, KS]). We conclude that the map f → v f mod constants injects H 1,p (M ) into the space of L p functions on the sphere (mod constants). In particular, H 1,p (M ) is Hausdorff, and T 1,p (M ) = 0 (according to 2.2, this is equivalent to a linear isoperimetric inequality originally due to S. T. Yau, [Y1]).

Vanishing of cohomology

Since every function with gradient in L p is modelled on a radial function, to prove vanishing of cohomology, one merely needs check wether radial extensions have their gradient in L p or not. For negatively curved homogeneous spaces [H], this happens if and only if p > p(M ) = tr(α)/ min ℜe(sp (α)), where M is identified with a semi-direct product N × α R defined by a derivation α whose eigenvalues have positive real parts. See [P1] for details.

Conformal dimension

We give a very brief account of the idea of conformal dimension of a hyperbolic group. Precise definitions and statements should be sought in the primary litterature, see for example [BP] and references therein.

A geodesic metric space is hyperbolic if it looks like a tree : there exists a constant δ such that all triangles are δ-thin, i.e., in a geodesic triangle, any point on one side is at distance at most δ of a point of one of the other two sides. Trees, of course, and hyperbolic spaces over R, C, H and O are typical examples.

Such metric spaces share many properties of hyperbolic spaces. They have a functorial compactification obtained by adding an ideal boundary. Depending on the choice of an origin, a metric is defined on the ideal boundary. Changing the origin changes the metric to an equivalent one.

A finitely generated group, once a finite generating set is chosen, becomes a metric space. The group is said to be hyperbolic if it is so as a metric space. If so, the boundary metric has a well defined Hausdorff dimension Q and Hausdorff measure µ, and the measure of a ball of radius r is proportionnal to r Q . One says that the metric is Ahlfors-Q-regular. Under a change of generating set, the compactification does not change, but the metric on the ideal boundary does. It changes to a quasiconformally equivalent one. Roughly speaking, this means that the two metrics have comparable balls, but without any control on their radii. In general, the Hausdorff dimension changes. In order to obtain a well defined numerical invariant of hyperbolic groups, one defines conformal dimension Cdim as the infimum of dimensions of metrics on the ideal boundary in the natural quasiconformal class (technical definitions vary, each of [START_REF] Pansu | Dimension conforme et sphère à l'infini des variétés à courbure négative[END_REF][START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF][START_REF] Keith | Conformal Assouad dimension and modulus[END_REF][START_REF] Bonk | dimension and hyperbolic groups with 2-sphere modulus[END_REF] uses a different definition).

Dimension minimizing metrics

The issue of wether this infimum is achieved or not is quite interesting. Based on work by S. Keith and T. Laakso, [KL], M. Bonk and B. Kleiner [BK] show that Ahlfors Q-regular metrics which minimize dimension in their quasiconformal class are Loewner. Roughly speaking, the Loewner property (introduced in [HK]) means that there are enough rectifiable curves so that Q-capacities of condensers behave like in Euclidean space. Boundaries of negatively curved homogeneous spaces and Fuchsian buildings admit Loewner quasiconformal metrics.

G. Elek [E1] and then M. Bourdon and H. Pajot [BP] have extended to hyperbolic groups the discussion of 5.2. L p cohomology of the group becomes a function space B p on the ideal boundary. If there exists a quasiconformal metric on the boundary with Hausdorff dimension equal to Q, then B Q = 0. This implies that the critical exponent for L p cohomology satisfies p ≤ Cdim.

On an Ahlfors Q-regular Loewner space, all possible notions of dimensions, including the critical exponent for L p cohomology, coincide, [Ty, BP]. Therefore hyperbolic groups which admit dimension minimizing Ahlforsregular quasiconformal metrics must satisfy p = Cdim. M. Bourdon and H. Pajot construct hyperbolic amalgamations Γ = A * C B such that

Cdim(Γ) ≥ min{Cdim(A), Cdim(B)} > 2, but L 2 b 1 (Γ) ≥ L 2 b 1 (A) + L 2 b 1 (B) -L 2 b 1 (C) > 0.
Then p(Γ) ≤ 2 < Cdim(Γ), ruining our hope of finding an optimal metric on the boundary of such a group.

Conclusion

The main point in this section is that, for simply connected negatively curved manifolds and groups, H 1,p comes from the ideal boundary.

We interpret this fact as follows. Canonical deformation retractions, and specificly their variants where the vertex sits at infinity, can be used in reverse : points are moved away from the vertex instead of towards the vertex. Instead of a vanishing theorem (cohomology of space equals cohomology of point), one gets that cohomology of space equals cohomology of ideal boundary.

Curvature pinching

In this section, canonical deformation retractions and their reversed forms will be used systematicly to investigate L p -cohomology in higher dimensions.

Horospherical coordinates

Horospherical coordinates are polar coordinates centered at a point at infinity. Such coordinates (normal exponential of a horosphere H) provide a diffeomorphism of M onto H × R, in which the metric reads again dr 2 + g r with simpler estimates on the growth of g r , e 2r √ -δ g 0 ≤ g r ≤ e 2e g 0 , and the constructions of 5.2 carry over verbatim.

Denote by ξ = ∂ ∂r the unit vector field tangent to the normals of H, and by φ t its flow. The radial limit in horospherical coordinates of a function f is simply u f = lim t→+∞ f • φ t . The conclusion of the discussion in 5.2 can be formulated as follows. For all p, if ω = df is a closed L p 1-form, then the limit u ω = du f = lim t→+∞ φ * t ω ∈ L p -1 (H) exists, and provides an injection of H 1,p (M ) into the space of exact 1-forms on H with coefficients in L p -1 . Therefore H 1,p (M ) is Hausdorff. It is in this form that the method extends to higher degrees.

Boundary values for differential forms : constant curvature

Let ω be an L p k-form on H n R . To prove that the limit lim t→±∞ φ * t ω exists, write

φ * t ω -ω = d t 0 φ * s ι ξ ω ds.
Since φ t expands by a factor at most e t , it expands k -1-forms by at most e (k-1)t . Since φ t expands volumes by at least e (n-1)t , L p -norms of k-1-forms are multiplied by at most e

(k-1-n-1 p )t . If p < n-1
k-1 , the integrand decreases exponentially, and the integral converges up to +∞. This shows simultaneously that the limit ω(+∞) exists, and that ω is exact if ω(+∞) vanishes, and complete the proof that H k,p (M ) is Hausdorff. ω(+∞) can be interpreted as a boundary value. If furthermore p ≤ n-1 k , the L p norm of k-forms is contracted by the reversed canonical deformation retraction, so that ω(+∞) = 0 and H k,p (M ) = 0. On the contrary, if n-1 k < p < n-1 k-1 , every closed L p k-form on the boundary is the boundary value of some L p -cohomology class, i.e. H k,p (M ) = 0.

If p > n-1 k-1 , the integrand increases exponentially, and the integral converges up to -∞. But the L p norm of k-forms is expanded by the direct canonical deformation retraction, thus ω(-∞) vanishes, and ω is exact. In other words, H k,p (M ) = 0.

If p = n-1 k-1 , none of the methods applies. Poincaré duality implies that reduced L p -cohomology vanishes. It turns out that torsion is nonzero. These results are summed up in the following picture. Let ω be an L p k-form on M . To prove that the limit lim t→+∞ φ * t ω exists, write again

φ * t ω -ω = d t 0 φ * s ι ξ ω ds.
Again, since φ t expands by a factor at most e t , it expands k -1-forms by at most e (k-1)t . Since φ t expands volumes by at least e (n-1) √ -δt , L p -norms of k -1-forms are multiplied by at most e

(k-1-n-1 p √ -δ)t . If p < n-1 k-1
√ -δ, the integrand decreases exponentially, and the integral converges up to +∞. As before, this shows that H k,p (M ) is Hausdorff.

Note that the obtained upper bound n-1 k-1 √ -δ is not the one stated in Theorem 3. A more careful inspection of the norm of the operator φ * t on k -1-form-valued 1 p -densities is needed to obtain the sharp bound. Homogeneous spaces for which Theorem 3 is sharp are obtained as semidirect products of R n-1 with R defined by diagonal matrices α with exactly two distinct eigenvalues. For instance, for n = 4, δ = -1 4 and k = 2, the matrix α = diag(1, 1, 2) does the job.

The same matrix defines a derivation of the Heisenberg Lie algebra span(X, Y, Z) with relator [X, Y ] = Z. The corresponding semi-direct product S = Heis × α R admits a left-invariant metric which makes it isometric to complex hyperbolic plane H 2 C . Nevertheless, Theorem 3 is not sharp for 2-forms on H 2 C , as we shall see next.

Critical exponents

Let ξ be a left-invariant vectorfield on a Lie group G, with flow φ t . For a given degree k ≤ dim(G), say an exponent p is critical in degree k if there exists an eigenvalue of Λ k ad ξ whose real part is equal to tr(ad ξ )/p. For instance, for ξ = ∂ ∂r on the semi-direct product S isometric to H 2 C , in degree 1, there are 2 critical exponents, 2 and 4.

Below critical exponents, the flow φ t contracts L p norms of k-forms, above critical exponents, φ -t contracts them. In these ranges, the method of 6.3 applies and yields vanishing of torsion T k+1,p . And indeed, for H 2 C , Theorem 3 predicts that T 2,p = 0 for p < 2 and for p > 4. However, one can prove more.

Theorem 5 ([P5]). T 2,p (H m C ) = 0 for m ≤ p < 2m.
To alleviate notation, we shall treat only the 2-dimensional case.

Two-sided boundary values

The trick is to split each differential form ω into its contracted part ω -and expanded part ω + , and treat each part separately. In other words, when p is not critical in degree k -1, the operator

B : ω → 0 -∞ (φ t ) * ι ξ ω -dt - +∞ 0 (φ t ) * ι ξ ω + dt,
is bounded on L p k-forms. When p is critical for no degree, P = 1-dB -Bd defines a chain homotopy of Ω * ,p to the complex B * ,p of differential forms on G

1. whose components have -1 derivatives in L p ; 2. which are killed by ι ξ and ι ξ d.

Although the Besov complex B * ,p is defined in a rather implicit manner, its cohomology turns out to be computable in some cases.

The Besov complex for H n

R

The critical exponents for H n R are numbers of the form n-1 k-1 , for k = 2, . . . , n-1. When p ∈ ( n-1 k , n-1 k-1 ), B * ,p vanishes in all degrees but k. This shows that H * ,p is Hausdorff, is nonzero only in degree k and identifies with B k,p . In that degree, it consists of differential k-forms whose coefficients belong to the Besov space B -k+ n-1 p p,p (S n-1 ), see [T].

6.7 The Besov complex for H 2 C Elements of the Besov complex can be viewed as differential forms on the Heisenberg group Heis with certain components missing. Denote by (dx, dy, τ ) the basis of left-invariant 1-forms on Heis, dual to (X, Y, Z). For 2 < p < 4, a typical element of B 1,p can be written e = f τ where f is a distribution on the Heisenberg group. Then

de = df ∧ τ + f dτ , f = -de(X ∧ Y ) .
One shows that there exist a constant c such that e B p ≤ c de B p .

This implies that the image dB 1,p ⊂ B 2,p is closed. Therefore T 2,p = 0 for 2 < p < 4. This completes the proof of Theorem 5.

Non vanishing of cohomology for H 2

C

One uses Poincaré duality, as formulated in [GT]. Let p ′ = p/p-1 denote the conjugate exponent. Let ω ∈ Ω k,p be a closed form. If there exists a closed form ψ ∈ Ω n-k,p ′ such that M ω ∧ ψ = 0, then the reduced cohomology class of ω does not vanish. Otherwise, one needs to construct a sequence ψ j ∈ Ω n-k,p ′ such that dψ j tends to zero but M ω ∧ ψ does not tend to zero.

In this way, one can show that reduced cohomology is non zero in open intervals, and that torsion is non zero in two of the critical cases, T 2,4 and T 3,4/3 . The remaining critical case is k = p = 2. For this, one appeals to Theorem 1 or [G2].

The known facts concerning the L p -cohomology of H 2 C are collected in the following table. Note that although p = 2 is a critical exponent in degree 1, nothing special seems to occur at p = 2 on 2-forms. This is confirmed by N. Lohoué, [CL, Lo]: L p -cohomology varies continously in p around p = 2. The known facts concerning the L p -cohomology of the -1/4-pinched Lie group R ⋉ diag(1,1,2) R 3 are collected in the following table. It that case, the situation at p = 2 is not fully understood.

Conclusion

All our L p -cohomology calculations are based on the fact that simply connected negatively curved manifolds deformation retract onto a horosphere. The result depends on the contraction properties of this retraction, which in turn depend on the exponent p. For p large (in degree 1, this never applies), the horosphere has to be viewed as a single point at infinity. For p small (in degree 1, this applies to all p < +∞), it has to be viewed as the whole ideal boundary. For intermediate, noncritical values of p, the retraction is used in both directions simultaneously, which is hard to interpret geometrically.

Proof.

  Let κ be a reduced L 2 -cohomology class, represented by a closed L 2 i-form α. Since the Kähler form ω is bounded, Corollary 9 provides us with a bounded 1-form β such that ω = dβ. Then ω ∧ α = d(β ∧ α), and β ∧ α ∈ L 2 , showing that the reduced cohomology class [ω ∧ α] = [ω] ∧ κ vanishes. If i < m, Corollary 8 implies that κ = 0. This shows that R i,2 ( M ) = 0.Sincedα 2 = d * dα, α ,vanishing of torsion is equivalent to a positive lower bound for the spectrum of the operator d * d, orthogonally to its kernel, and follows from a positive lower bound for the spectrum of ∆ = d * d + dd * orthogonally to its own kernel.

  values for differential forms : pinched curvature Assume M has δ-pinched curvature.

  Laplacian on i-forms is bounded below if i > m. Using the Hodge * -operator, one gets the bound for i < m.

		This
	gives estimates
	α ≤ const. dα ,	α ≤ const. d

* α , which show that d and d * have a closed image. In particular, dL 2 Ω m-1 (M ) and d * L 2 Ω m+1 (M ) are closed. Since ∆ commutes with d and d * , one gets a spectrum bound on exact and coexact m-forms as well. Using the Hodge decomposition