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Tree-Structured Lattice Vector Quantization

Vincent Ricordel, Claude Labit

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Vector Quantization by Packing of Embedded Truncated Lattices

Vincent Ricordel, Claude Labit

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Matthieu Perreira Da Silva

Vincent Courboulay

Armelle Prigent

Matthieu Perreira
email: mperreir@univ-lr.fr

Da Silva

P Estraillier

Da Silva L3i

Fast, low resource, head detection and tracking for interactive applications Fast, low resource, head detection and tracking for interactive applications

Keywords: Head-based interaction, gameplay, low resource, low cost, head tracking

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Introduction

Providing good human-computer interactions is an important point in the development of interactive software [START_REF] Barr | Video game values: Human-computer interaction and games[END_REF]. In the field of games, it is a way to increase immersion in the virtual world of the player. Classical interactions with mouse, keyboard or gamepad, are limited in comparison with the reality of graphics displayed. Indeed, a growing trend is to concentrate on new kinds of interfaces between the player and the virtual world. For example, some approaches are using a headpiece device to detect head movements in order to change the game camera direction. Another example is the use of the interactions between human bodies as a core gameplay element ("Teaser of the EPIDEMIK exhibition -Cité des sciences et de l'industrie (Paris, France)," pas de date). In this paper, we describe a This is a draft of the final paper that can be retreived from : http://www.psychnology.org/ fast low cost head tracking system which is used to experiment new kinds of gameplay and improve the player's experience.

We present four prototype applications that have been developed or modified in order to use head direction as input. These applications have an increasing level of integration of head tracking in the gameplay. The first two are a Tetris and a racing game where head direction is simply used as an input to directly control the game. The third application is an adventure game in which user's head direction helps modifying the game scenario in order to try to make the game more immersive and fun to play.

The last one is a pedagogical game in which information about the user's attention (derived from head direction) is used to adapt the game unfolding in order to refocus the player's attention. It is used as a tool for pedo-psychiatrists working with children with autism in the pedo-psychiatric hospital of La Rochelle, targeting the objective of improving children's attention.

In chapter 2, we describe the low cost head tracking system that we have developed.

In chapter 3, we show how this system can be used as an explicit way to control a game. Then in chapter 4, we describe a more advanced software architecture that uses gaze tracking in order to detect the players' attentional state and update the game unfolding accordingly.

Low cost, fast head tracking

Many face detection algorithms have been published over the last 15 years. A survey of many of these systems can be found in [START_REF] Yang | Detecting Faces in Images: A Survey[END_REF] and [START_REF] Hjelmås | Face Detection: A Survey[END_REF]. Face or head tracking is also a very active research area. See for example (Erik Murphy-Chutorian & Mohan Manubhai Trivedi, 2009) for a recent survey of the existing systems. However, many of these systems are conceived for performing either face detection or tracking and are not real-time capable. Recent work (E. Murphy-Chutorian & M.M. Trivedi, 2008) proposes a robust and real time system (30fps) for head detection and tracking, but this system uses the GPU in order to be real time. As a consequence, the system uses most of the CPU and GPU resources of the computer and no interactive application can be run in parallel.

To overcome these limitations, we propose a low resource head detection and tracking algorithm which allows running an interactive application (even a 3D game) in parallel on the same computer.

Constraints

As our system is designed to be used by a wide range of applications and users (from educational games for children with autism to adventure games for "common gamers"), some constraints have emerged:

-non invasive material: users should be able to forget the presence of the tracking device and concentrate on the interactive application; -low cost: the system must use affordable, off the shelf, hardware; -single user: the system should detect and track only one person at a time. If multiple persons are present in front of the camera, only one of them will be tracked.

-recordable information: the algorithm should output tracking information that can be easily stored or exploited by another algorithm (ex: attention model).

-standard computer: the system should run smoothly on a standard mid-end computer, with no additional or specific hardware.

-unconstrained environment: our algorithm should be able to work without any additional lighting.

The system is based on a low cost (ieee-1394 or USB) webcam connected to a standard computer. Despite its low cost, this type of camera captures video frames of size 640x480 at 30 frames per second which are suitable characteristics for both accurate face features localization and efficient face features tracking. Depending on the application, the system can use a color or grayscale camera. The choice of a grayscale camera instead of a more common color camera will be driven by the environment of the aimed application. Indeed, in some environments (mainly indoor), the amount of light available is often quite low. As grayscale cameras usually have better sensitivity and have a higher image quality (as they don't use Bayer filters) they are more suitable for these environments. Grayscale cameras may also be used with infra-red light and optics that don't have infra-red coating in order to improve the tracking performance by the use of a non invasive more frontal and uniform lightning.

During our experiments we have only used (so far) color or gray scale cameras with no additional infrared lighting.

Architecture

The tracking algorithm is built upon four modules which interoperate together in order to provide a fast and robust face tracking system. The algorithm contains two branches: the first one for face detection and the second for face tracking. At run time, the choice between the two branches is made according to a confidence threshold, evaluated in the face checking and localization module. In the following paragraphs, we detail the algorithms used by each modules of Figure 1.

Pre-processings

Before face or radial symmetry detection, the input image must be pre-processed in order to improve face and face feature detection performance. The main preprocessing steps are:

-image rescaling; the input image is rescaled so that low resolution data is available for radial symmetry detection algorithms; -(in case of color camera) skin hue detection, using a commonly used histogram-back-projection [START_REF] Zaqout | Pixel-based skin color detection technique[END_REF] -lighting correction, also called contrast normalization which consist in adapting each pixel intensity according to the local mean intensity of its surrounding pixel. As this task is naturally performed by human's retina, several complex models have been developed to mimic this processing [START_REF] Beaudot | The neural information in the vertebra retina: a melting pot of ideas for artifical vision[END_REF].

Since our system needs to be real-time, we have chosen to approximate this retinal processing by a very simple model which consists in the following steps:

-For each image pixel, we build a weighted mean M x,y of its surrounding pixels. In our implementation we used first order integral filtering (Bartlett Filter) in order to achieve fast filtering. Note that first order integral filters are an extension of the commonly used zero order integral filters (box filters). For more information about generalized integral images see [START_REF] Derpanis | Fast scale-space feature representation by generalized integral images[END_REF].

-Then we calculate the normalized pixel intensity:

Radial symmetry detection for face shape and eye detection

Once the image is pre-processed, we use a set of radial symmetry detectors in order to localize a candidate face region that will be further checked by the face checking and localization module. Once again our real-time constraint guided the choice of the algorithms we used.

Face ovoid shape is detected in a low resolution version of the input image (typically 160x120, since we don't need much precision for this step) using an optimized version of the Hough transform (Figure 2.c) whereas eyes are detected using an optimized version of the Loy and Zelinsky transform (Loy & Zelinsky, 2003) (Figure 2.d). In order to speed up both transforms, the following improvements have been made: only pixels with a gradient magnitude above a pre-defined threshold 1 are processed; the algorithms vote in only one accumulator for all radius and accumulators are smoothed only once at the end of the processing.

Using these two symmetry maps and a set of face geometry based rules, we define the face candidate area.

1 defined empirically

Face checking and localization

This module serves two purposes:

-When called from the face detection branch, it checks if the face candidate area really contains a face and outputs the precise localization of this face.

-In the case of face tracking, it only finds the new position of the face.

In both cases, the module outputs a confidence value which reflects the similarity between the interface face model and the real face image.

Face checking and localization is based on the segmentation of the face candidate image into several blobs, determined as follow: the source frame is first filtered by a difference of Gaussian (DoG) filter; the result image is then adaptively thresholded.

The resulting connected components (blobs) are then matched on a simple 2D face model in order to check and localize the face.

Once the face is localized, the 2D face model is deformed to adapt to the detected face features. These deformations are kept from one tracking step to the other in order to improve tracking performance. Since we are performing continuous tracking at 30 fps, changes from one image to the other are quite small, hence we do not use any king of prediction (ex: Kalman filtering).

Head pose estimation.

Similarly to previous research we use triangle geometry for modeling a generic face model. For example, [START_REF] Kaminski | Head Orientation and Gaze Detection from a Single Image[END_REF] resolves equations derived from triangle geometry to compute head pose. Since we favor speed against accuracy, we use a faster and simpler direct approximation method based on side length ratios [START_REF] Nikolaidis | Facial feature extraction and pose determination[END_REF].

a b c h f E F G a b c h f E F G h E F G u v H h E F G u v H Figure 3
. Features used for head pose estimation.

Fast, low resource, head detection and tracking for interactive applications Continuous values for pan, tilt, roll as well as x, y and z position are processed using simple triangle geometry. The considered EFG triangle is built from the respective position of left eye, right eye and nose tip. We hypothesize that for a frontal view, this triangle is isosceles.

2 3 a h et c b a f = = =
When the face is rotated we can determine the following values easily: with

Performance and robustness

Processing time

We measured the processing time of the algorithm on a laptop PC equipped with a 1,83GHz Intel Core Duo processor. We obtained the following mean processing times:

Algorithm Processing time per frame CPU load

Face detection 2 30 ms 50% (100% of one core)

Face tracking 16 ms 25% (50% of one core)

Table 1. Processing time of the detection / tracking algorithms.

Processing time include all processing steps, from pre-processing to head pose estimation. Since image capture is done at 30fps and the algorithm is using only one of the two processor cores the algorithm is fast enough to enable running a game in parallel on a standard middle-end computer. Please note that we have not used any GPU acceleration in order not to slow down any 3D graphics in the target interactive application.

Robustness

As can be seen from the tracking example shown on Figure 4, the algorithm can handle a broad range of head orientation and distance, as well as different people. The However, this algorithm is designed to be fast, as a consequence tracking performances still need to be improved under some lightning conditions (backlit scene, directional lighting casting hard shadows, etc.) and head pose (Figure 6). When using a color camera and skin color detection algorithms (as in [START_REF] Schwerdt | Robust Face Tracking Using Color[END_REF] or [START_REF] Séguier | A very fast adaptive face detection system[END_REF])) the face detection and tracking robustness are improved. This modification however prevents us from using infra-red light to improve the algorithm performances under poor lightning conditions. A possibility would be to add an infrared lightning and adapt the current algorithm to this new lightning in order improve the robustness of the system. Lastly, the algorithm is not designed for occlusions handling: if the face is masked by any object, tracking is lost (but quickly recovered at the end of occlusion). Fortunately, occlusions are very rare when the system is used with interactive applications.

Gaze tracking

Some of the applications described below require information about user's gaze. Since eyes are a feature extracted for our face model, it should be quite easy to use their position to (in conjunction with head orientation) estimate eye gaze. However, considering the large freedom of movement allowed to users, extracted faces can be as small as 48 x 57 pixels. In this case, it is difficult to compute an accurate and reliable estimate of gaze direction.

To overcome these problems, we propose to use head direction as an approximation for gaze direction. Although it may look simplistic, in practice, it provides enough information for the simple attention monitoring tasks described in chapter 4.

The algorithm described above has been integrated into several games / interactive applications. In the next chapter we describe the experiments conducted on these applications.

Explicit control of games via head tracking

Head as a simple input device

Our first experiments consisted in modifying two classic games: a Tetris and a racing game (which are both public domain Microsoft XNA Game Studio game samples) in order to connect them with our head tracking system. We chose these games because any casual gamer knows how to play them, which shortens the learning phase.

Moreover, racing game players tend to naturally roll their head when the car is within curves; head movements seemed therefore good candidates for an alternative way to control the car. Once the games modified, we were able to use head movements like any other input device.

Figure 7 show screenshots of both applications. Video demonstrations are also available on YouTube 3 .

We conducted some experiments with a small set of five persons aged from 23 to 35 (all were casual gamers). Each person was shown how to play the game for 5 minutes.

Then, they could play on their own for 15 more minutes. Lastly, they were interviewed in order to know what they thought about this new kind of interaction. These experiments showed that:

-Gaze control is not suited to all kinds of games. Moving and rotating Tetris bloc with head movements is quite unintuitive and requires some training. On the contrary driving with gaze is quite easy to learn, and quite fun.

-Robustness is a key point for a good gaming experience. If tracking is lost during the game (because of extreme head positions or bad lightning condition as described in section 2.3), all immersion improvements are lost. However, the system need also to be fast, if the system's responsiveness is low immersion improvements are also lost.

-The gameplay of head controlled games needs to be adapted in order to be really intuitive. For example, gamers found it difficult to control the Tetris game with their head. Additionally, they obtained lower scores when controlling the game with their head than when controlling it using the keyboard. For the racing game, although head control was easier and fun, users sometimes had difficulties controlling throttle. A solution would be for example, mixing mouse (throttle) and head tracking (steering) in order to propose a more natural gameplay. To further exploit the potential of gaze in games we have decided to add head controlled sequences in an adventure game prototype developed by students of the ENJMIN 4 School: L3iLife. This time we don't replace an input modality by another.

Instead, we introduce new gameplay elements which were created for head based interaction.

Head as a new gameplay element

We have developed an adventure game based on the visit of a virtual world that represents the computer science laboratory (L3i) of the University of La Rochelle. The player has to explore the laboratory by opening doors and visiting rooms. Figure 8 shows a screen capture of the prototype that has been developed with the Unreal Engine editor5 .

The game concept is the following one. The player is a student that has a fixed delay to give a work to his teacher. He is in direct competition with an evil student that tries to prevent the player from reaching his goal and with a little pest that tries to steal his work and give her own work first to the teacher.

We propose to give to the player a maximum amount of interactivity while keeping a robust and interesting narrative framework. The approach of emergent narrative consists of a particular architecture that increases the player's freedom of action and produces a dynamic control of narrative quality. A challenge is, for example, to detect the player's behavior in order to dynamically modify the scenario.

Interested readers will find more details about the architecture of the game in

Game / head interaction

During our first experiments (conducted with the same protocol and participants as in section 3.1), we only took into account a few explicit players' behavior:

Firstly, we focused on the interaction with the non player character of the little pest.

She tries to steal the work of the player. The player can interact with the girl by doing a wink6 to the camera when he is in front of the girl. Then, the girl will give his work back to the player if she has stolen it to him.

Secondly, we allow the player to protect himself against the evil student by looking down. Indeed, if the evil student arrives near the player, this one will put his head down and then the evil student will go on without stealing his work. During our first experiments we have observed that this new kind of interaction improves the players' immersion in the virtual world of the game. Moreover, it also increases its interest for the game because the gameplay is richer and the game more fun to play. These observations will certainly be confirmed once our framework will include the observation of implicit behavior as we will have more real-time feedback on the user's gaming experience.

We are currently working on the integration of more complex kinds of behavior into the interactive game framework. Gaze tracking could be used to observe the level of attention of the player. For example, if the player stops watching at the screen, a particular game action can be launched to refocus his attention. In L3iLife for example, in this kind of situation, the adaptive architecture can modify the unfolding of events, and make the evil student run after the player to steal his work. It is a stressing action that can bring some interest back to the player. This implicit behavior system is not yet implemented in L3iLife, but we conducted some similar experiments with an educational game aimed at autistic children. We describe this system in the next section.

Implicit control of games via head tracking

Usually, computer software has a very low understanding of what the user is actually doing in front of the screen. It can only collect information from its mouse and keyboard inputs. It has no idea of whether the user is focused on what it is displaying, it doesn't even know if the user is in front of the screen.

One of the first steps to making computer software more context aware is to make it attention aware. In [START_REF] Roda | Attention Systems: Theories, Applications, and Research Agenda[END_REF] attention aware systems are defined as «Systems capable of supporting human attentional processes.» From a functional point of view, it means that these systems are thought as being able to provide information to the user according to his estimated attentional state.

According to [START_REF] Roda | Attention Systems: Theories, Applications, and Research Agenda[END_REF], one of the best ways to estimate user's attention is using gaze direction. Consequently, building good adaptive attention aware system requires estimating reliably user's gaze.

Many high accuracy commercial gaze tracking systems are currently available (Abdallahi Ould Mohamed, Perreira Da Silva, & Vincent Courboulay, 2007). Most of them use dedicated or costly hardware in order to estimate user's gaze with the desired precision. Moreover, these systems are designed to be used with cooperative users (most of the time disabled users or attention studies participants) in a constrained environment. A choice needs to be made between gaze tracking precision, equipment cost and user's freedom of movement. As our attention aware system only needs to know whether the user watches the screen or not, we propose to use head pose as an estimator for gaze direction.

But before closing this, let's have a closer look on what attention is.

Attention

Attention is historically defined as follows [START_REF] James | The Principles of Psychology[END_REF]: It has been proven that the same functional brain areas were activated for attention and eye movements [START_REF] Corbetta | A common network of functional areas for attention and eye movements[END_REF]. Consequently, the best attention marker we can measure is undoubtedly eyes and gaze behavior. A major indicator concerning another type of attention, vigilance, named PERCLOS [START_REF] Dinges | Evaluation of techniques for ocular measurement as an index of fatigue and the basis for alertness management[END_REF] is also using such markers. [START_REF] Horvitz | Models of attention in computing and communication: from principles to applications[END_REF] presents different models for integrating attention into an Attentional User Interface. But the models are mainly focus on attention, without any clear link with gaze. On the contrary, [START_REF] Peters | A model of attention and interest using Gaze behaviour[END_REF], presents how gaze direction can be used in order to improve the behavioural plausibility of an Embodied Conversational Agent. But the models described have only been used in a simulated environment.

«
In continuity of such studies, we based our markers on gaze behavior, approximated by head pose, to determine selective and maintained attention. A weak hypothesis is that a person involved in an interesting task focuses his/her eyes on the salient aspect of the application (screen, avatar, car, enemy, text...) and directs his/her face to the output device of the interactive application (screen). Nevertheless, if a person does not watch the screen, it does not necessarily mean that he/she is inattentive; he/she can be speaking with someone else about the content of the screen [START_REF] Kaplan | The challenges of joint attention[END_REF]. However, the more time users spend not watching the screen, the more probable is their inattention. Consequently, we have decided to adopt the following solution: if the user does not watch the screen during a time t, we conclude to inattention. In the following subsection, we present how t is determined. If inattention is detected, we inform the application.

A simple model of human inattention

The goal of this model is to define the delay after which the application tries to refocus the user on the activity. Actually, we easily understand that in this case, an interactive application does not have to react the same way if people play chess, role player game or a car race. Until now, this aspect of the game was only directed by the time during which nothing was done on the paddle or the keyboard.

We based our model of what could be named inattention on two parameters:

-the type of application;

-time spent using the application.

The last parameter depends itself on two factors:

-a natural tiredness after a long time -a disinterest more frequent during the very first moments spent using the application than once attention is focused, this time corresponds to the delay of immersion.

Once the parameters are defined, we propose the following model in order to define the time after which the application try to refocus the player who does not look at the screen.

Potential of attention

As we mentioned, potential of attention depends mainly on two parameters, tiredness and involvement. We have decided to model arousal (the opposite of tiredness), or potential of attention, by a sigmoid curve parameterized by a couple of real number β1 and β2. β2 represents the delay after which the first signs of fatigue will appear and β1 is correlated to the speed of apparition of tiredness (Figure 10).

2 1 2 1 exp 1 exp β β β β β β β β β β β β β β β β + + + + - -- - + + + + - -- - + + + + = = = = t t arousal P
where β 1 and β 2 are two real parameters.

For the second parameter, we have once again modeled involvement, or interest probability, by a sigmoid. We started from the fact that activity is a priori fairly interesting, but if the person is involved after a certain time ruled by α2, we can consider that interest is appearing at a speed correlated to α1 (Figure 10).

Delay of inattention

Once this model is defined, we are able to determine the time after which the software has to react (if the person still does not look at the screen). Here, it is an arbitrary threshold γ determined by experience, which characterizes each application.

The more the application requires attention, the higher this coefficient is. Under the hypothesis that attention follows an exponential decay, the equation we have adopted is an exponential function which models the time after which attention reaches its activity switching threshold. As a conclusion, we can summarize our model of inattention by the two following steps (Figure 11):

-depending on the time elapsed from the beginning of the application, we estimate the potential of attention P attention (t); -depending on this potential and the application, we estimate the delay D gamma(t) (P attention (t)) after which the software has to refocus the inattentive player.

Please note that this model was validated on only five children and still need to be validated on more people. Nevertheless, the first tests are quite promising. This model of inattention has been used in conjunction with our real time gaze tracking system in an educational game. This game is part of a project called the AutiSTIC Project which we describe in the following section.

The AutiSTIC Project

The AutiSTIC Project tends towards implementing a system that can help autistic children during the rehabilitation process. The role of such a system is to provide the child with personalized activities in the form of educational games. During a session, the system collects through various devices (camera, touch screen, mouse, and keyboard) actions and attention, in order to understand her/his behaviour and responds to it, in real time, by adequate actions considering directives. These directives concern rupture, avoidance, stereotype gestures... For instance, the system may attract attention by displaying an image on the screen, or by launching a characteristic music. It is impossible to generalize activities or reaction without precaution; we have to favour the adaptability of a system to take into account the specificity of persons. It is important to locate and interpret carefully these intrinsic behaviours as eye and gaze orientation, in order to help him/her to rehabilitate. We do not have to perturb the child when he/she is working; misinterpreting his/her attention may deeply perturb him/her because of his/her monotropsim [START_REF] Murray | Attention, monotropism and the diagnostic criteria for autism[END_REF].

In the application context, our architecture aims to bring flexibility and modularity in the individualized rehabilitation of children with autism. In the next section we present a part of our platform which observes children and analyses their visual attention.

The hello hidden game

This interactive game, which was developed within the context of the AutiSTIC project, is characterized by simplicity in order to not perturb the autistic child, often sensitive to complex environments. This is why the game has a static background, contains few objects and is easy to use. Nevertheless, each object has several behaviors in order to allow the control of execution according to the behavior of each child. The goal of the game is to allow autistic children to reach by interactive manipulation the competences of perception, motor function, spatial and temporal representation. Figure 12 shows the game interface of "hello hidden". This game allows using one or more balls of various colors on the screen. These balls appear in a frame that is displayed in fullscreen.

There are two kinds of balls:

-The small ball, called cursor, which the children can handle by applying a pressure on a touch screen.

-Big balls of various colors which remain motionless. The interaction between the cursor ball and the big balls contains two possibilities: Either the cursor ball disappears when it becomes near big ball and reappears when it goes away, or it stops its progression when it arrives at the periphery of a big ball and joins with it.

The objectives of this game vary according to various situations which occur. In the case of the presence of only the cursor ball, its elementary use allows the child to establish a relation between a direct motor action (to touch the screen) and the effect produced (movement of the object). In the case where one or more big balls are present on the screen and the cursor ball disappears when it moves near big ball, the objective is to analyze the capacity of the child to represent hidden objects and to act to make the cursor visible. More details about the architecture of this interactive application can be found in (Karim Sehaba, Vincent Courboulay, & Pascal Estraillier, 2006). In this application, head orientation is used to estimate whether the child is watching the screen or not.

When he is not watching the screen, we use our inattention model to decide when to play a sound to refocus him.

Results

Our platform can use, in real time, information concerning the explicit behavior of the child (mouse and keyboard events) and his/her implicit behavior (movements of the face, gaze...) in order to estimate the degree of carelessness of the child and to adapt the activity as well as possible. We use these annotations, done using video annotation software (L3iAnnote), in order to make automatic measures and to validate our model of inattention. Our software, installed both in academic and in medical sites, has allowed us to confront our model and the reality of children in front of a computer.

Preliminary results obtained with children who took part to our study (three) are very interesting and promising, but the number of participants should be significantly increased and the model also needs to be validated on people without autism. Yet, it is very difficult to find children for the experimentations, because of the reserve of parents and the small number of children in the medical structure. Currently, we have tested our model on ten ordinary players in our laboratory, and the first results confirm those obtained with children with autism. We will continue the workshops with the same children by leaving this time the initiative of the stimuli to the computer, in order to analyze their behavior.

Conclusion

In this article we have presented a low resource real-time head tracking system. This system provides a fast and low cost solution for integrating approximated user's gaze into interactive applications. Its precision is not as good as gaze tracking devices and its robustness is not as good as slower head tracking systems. However, it provides a good tradeoff between cost, resource consumption, precision, robustness and freedom of movement. Ideally, one would like a more precise and robust system; but providing a precise system with great freedom of movement or a more robust one with low resource consumption are still unsolved issues.

This system has been successfully integrated into four different games: a Tetris game, a racing game, an adventure game (L3iLife) and an educational game part of the AutiSTIC Project. The experiments we conducted on these different games showed that even if using head as a simple input device for explicit game control can improve the player's immersion, its full potential can only be exploited when adapting or building new gameplay. This can be accomplished by the following ways:

-Do not try to use full head based control. Mix classical input devices and head control in order to provide the best ergonomics.

-Insert head controlled sequences in games only when necessary. For example in L3iLife, head controlled sequences were launched only in defined situations (interaction with non-player character).

-Do not change the way the game is controlled, but use head tracking as a gaze tracking approximation in order to monitor user's attentional state. These data can then be used to modify the game's unfolding.

Lastly, we have proposed a simple model of inattention, which provides an original solution for processing the attentional state of person from its eye or head tracking data.

Acknowledgments

This work has been partly funded by Orange Foundation and French Poitou-Charentes County.

Figure 1 .

 1 Figure 1. Architecture of the face tracking and pose estimation system.

  source image, I the normalized image, and A a normalization factor.

Figure 2

 2 Figure2shows the result of our simple contrast normalization algorithm on a side lit scene.

Figure 2 .

 2 Figure 2. a) Tracking result of a side lit scene. b) Source image after lightning correction. c) Result of face ovoid detection. d) Result of eyes detection.

  normalization step also allows the algorithm to run under different illumination conditions.

Figure 4 .

 4 Figure 4. Result of the tracking algorithm for different face distances and orientation.

Figure 5 .

 5 Figure 5. Result of the detection / tracking algorithm for different participants.

Figure 6 .

 6 Figure 6. Examples of detection / tracking failure.Left and middle : face pan and tilt are to important. Right : backlightning is too strong.

Figure 7 .

 7 Figure 7. Left: screenshot of the gaze controled Tetris game. Right screenshot of the gaze controled racing game.

(

  Champagnat, A. Prigent, & P. Estraillier, 2005) and (Perreira Da Silva, Vincent Courboulay, & Armelle Prigent, 2007).

Figure 8 .

 8 Figure 8. Left: A screen capture of the prototype game (L3iLife). Right: Overview of the narrative based game architecture

Figure 9 .

 9 Figure 9. Gaze controled sequence of L3iLife. Left: The evil student is about to catch the player.Middle: The player gaze down in order to show his submissiveness. Right: The evil student does not catch the player: game is not over!

  Everyone knows what attention is. It is the taking possession by the mind in clear and vivid form, of one out of what seem several simultaneously possible objects or trains of thought...It implies withdrawal from some things in order to deal effectively with others» Thus, attention is the cognitive process of selectively concentrating on one thing while ignoring other things. In spite of this single definition, there are several types of attention (A. Ould Mohamed, V. Courboulay, K. Sehaba, & Menard, 2006): awakening, selective attention, maintained attention, shared attention, internal or external absentmindedness and vigilance. For an interactive task, we are mainly interested in selective and maintained attention. The analysis of the first one allows knowing whether people are involved in the activity. The second one enables us to assess the success of the application.

  model of potential of attention, we couple both previous models in order to obtain:

γ

  is a function of time because we have estimated that it can exist several tempo in an application (intensive, stress, reflection, action ...).

Figure 10 .Figure 11 .

 1011 Figure 10. Left: model of tiredness evolution. Middle: model of interest probability. Right: model of potential of attention. Abscissa represents time in minutes. (α1=1, α2=3, β1=0.3, β2=12)

Figure 12 .

 12 Figure 12. Left: Screenshot of the "hello-hidden" game. Right: General architecture of the adaptive software platform.

http://www.unrealtechnology.com/

Since the player's eyes are already localized by our face tracking algorithm, blinks are detected by a simple frame subtraction.

The authors would also like to thank doctor Mr. D. Lambert Head of Department of Child Psychiatry of La Rochelle hospital (France) and his team, in particular: Mr. V.

Gabet for their useful advices regarding the rehabilitation methods dedicated to children with autism.


Yves Stalder
email: yves.stalder@math.univ-bpclermont.fr

FIXED POINT PROPERTIES IN THE SPACE OF MARKED GROUPS

Keywords: Uniform actions, ultralimits, Property (T), Property (FRA)

We explain, following Gromov, how to produce uniform isometric actions of groups starting from isometric actions without fixed point, using common ultralimits techniques. This gives in particular a simple proof of a result by Shalom: Kazhdan's property (T) defines an open subset in the space of marked finitely generated groups.

Introduction

In this expository note, we are interested in groups whose actions on some particular kind of spaces always have (global) fixed points.

Definition 1.1. Let G be a (discrete) group. We say that G has: -Serre's Property (FH), if any isometric G-action on an affine Hilbert space has a fixed point [HV89, Chap 4]; -Serre's Property (FA), if any G-action on a simplicial tree (by automorphisms and without inversion) has a fixed point [Ser77, Chap I.6]; -Property (FRA), if if any isometric G-action on a complete R-tree has a fixed point [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts[END_REF]Chap 6.b]. These definitions extend to topological groups: one has then to require the actions to be continuous. Such properties give information about the structure of the group G. Serre proved that a countable group has Property (FA) if and only if (i) it is finitely generated, (ii) it has no infinite cyclic quotient, and (iii) it is not an amalgam [START_REF] Serre | Arbres, amalgames, SL 2 . Société Mathématique de France[END_REF]Thm I.15]. Among locally compact, second countable groups, Guichardet and Delorme proved that Property (FH) is equivalent to Kazhdan's Property (T) [START_REF] Guichardet | Étude de la 1-cohomologie et de la topologie du dual pour les groupes de Lie à radical abélien[END_REF][START_REF] Delorme | 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations[END_REF]. Kazhdan groups are known to be compactly generated and to have a compact abelianization; see e.g. Chapter 1 in [START_REF] De La Harpe | La propriété (T ) de Kazhdan pour les groupes localement compacts[END_REF]. It is known that Property (FH) implies Property (FRA) 1 , which itself obviously implies Property (FA).

We are interested in the behavior of these properties in Grigorchuk's space of marked (finitely generated) groups (see Section 2 for definition). One main aim of this note is to provide a simple proof of the following result, which implies that any finitely generated Kazhdan group is a quotient of a finitely presented Kazhdan group: Theorem 1.2 (Shalom [START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]). Property (FH) defines an open subset in the space of marked groups.

Rather than just prove this result, our purpose is to indicate a general scheme, which gives a common strategy for proving Shalom's result and Theorems 1.3 and 1.4 below.

Theorem 1.3 (Korevaar-Schoen [START_REF] Korevaar | Global existence theorems for harmonic maps to non-locally compact spaces[END_REF], Shalom [START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]). A finitely generated group G has Property (FH) if and only if every isometric G-action on a Hilbert space almost has fixed points.

Theorem 1.4 (Culler-Morgan). Property (FRA) defines an open subset in the space of marked groups.

In fact, one deduces the latter Theorem from [CM87, Thm 4.5] by applying it to a free group F n . On the other hand, it is an open problem whether Property (FA) defines an open subset in the space of marked groups.

The general (simple) idea for the scheme is, starting from actions without (global) fixed points, to pass to a "limit" of the spaces to get uniform actions (that is, not almost having fixed points; see Section 2). It follows in fact the same strategy as in [Gro03, Sect 3.8.B] 2 , and Theorems 1.2 -1.4 are parcular cases of Gromov's results.

Theorem 1.5 (Gromov). Let (G k , S k ) k∈N be a sequence of marked groups converging to (G, S). If each group G k acts without fixed point on a (non-empty) complete metric space (X k , d k ), then G acts uniformly on some ultralimit of the spaces (X k , d k ).

The definition of ultralimits will be given in Section 3. Note that we allow to rescale the spaces before taking the limit (see Theorem 3.12 for a more precise statement). The idea to take "limits" of metric spaces is not new, even for such purposes. Asymptotic cones, introduced by Gromov in [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF] and defined rigorously in [START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF][START_REF] Van Den Dries | Gromov's theorem on groups of polynomial growth and elementary logic[END_REF] are a major particular case of ultralimits which is very useful in the study of metric spaces and groups, see e.g. [Dru] and references therein. Ultralimits appear explicitely in [KL97, BH99], for instance. Finally, let us mention that Korevaar and Schoen [START_REF] Korevaar | Global existence theorems for harmonic maps to non-locally compact spaces[END_REF] introduced limits of CAT(0) spaces (with another process) and proved in this context results of the same spirit as Theorem 1.5.

Section 2 gives the necessary preliminaries. In Section 3, we recall what ultralimits are and prove Theorem 1.5. Finally, Section 3 is devoted to applications to Properties (FH) and (FRA).

Acknowledgements. I would like to thank Benjamin Delay for having pointed out a mistake in the proof of Theorem 3.12 and for his suggestions about a first version of this paper. Thanks are also due to Yves de Cornulier for references [START_REF] De Cornulier | Fixed points and almost fixed points, after Gromov and V. Lafforgue[END_REF][START_REF] Gromov | Random walk in random groups[END_REF][START_REF] Lafforgue | Une remarque sur les fonctions conditionnellement de type négatif[END_REF], to Vincent Guirardel for a useful discussion about the results in this text, and to the referee for her/his valuable suggestions. Finally, I am particularly grateful to Nicolas Monod and Alain Valette for their very valuable advices and hints, and for their remarks about previous versions of the text.

Preliminaries

2.1. Terminology about group actions. In this note, every metric space is assumed to be non-empty and all groups considered are discrete ones. Let (X, d) be a metric space and let G be a group acting on it by isometries. The action is said to almost have fixed points if, for all ε > 0 and for all finite subset F ⊆ G, there exists x ∈ X such that d(g • x, x) < ε for all g ∈ F ; it is said to be uniform otherwise.

An action with a global fixed point almost has fixed points, but the converse strongly does not hold. Indeed, the following examples show that an action with almost fixed points can be metrically proper, that is such that for any x ∈ X and R > 0, the set {g ∈ G : d(x, gx) ≤ R} is finite.

Example 2.1 (on the hyperbolic plane). Let H 2 be the Poincaré upper half-plane and define ϕ : H 2 → H 2 by ϕ(z) = z + 1. This gives a Z-action by isometries on H 2 which is metrically proper and almost has fixed points.

2. This was pointed out to me by Yves de Cornulier, after this paper was accepted for publication and posted on the arXiv. There also exist good unpublished notes [Cor] on this topic.

Example 2.2 (on a Hilbert space). Let S be the shift operator on ℓ 2 Z (defined by (Sξ)(n) = ξ(n -1)) and δ 0 ∈ ℓ 2 Z the Dirac mass at 0. The affine map ξ → Sξ + δ 0 defines a Z-action by isometries on ℓ 2 Z which is metrically proper and almost has fixed points.

Remark 2.3. In the case of finitely generated groups, the definition can be made easier: if S is a finite generating set of G, a G-action by isometries on X almost has fixed points if and only if, for all ε > 0, there exists x ∈ X such that max{d(x, s • x) : s ∈ S} < ε.

2.2. The space of marked groups. Let us recall that a marked group on n generators is a pair (G, S) where G is a group and S ∈ G n generates G. A marked group (G, S) defines canonically a quotient φ : F n → G, and vice-versa. Moreover, for such a quotient, we may consider the normal subgroup N = ker(φ) ⊳ F n . Two marked groups, or two quotients, are said to be equivalent if they define the same normal subgroup of F n . Abusing terminology, we denote by G n the set of (equivalence classes of) marked groups on n generators, or the set of (equivalence classes of) quotients of F n , or the set of normal subgroups of F n .

We now describe Grigorchuk's topology on G n [START_REF] Grigorchuk | Degrees of growth of finitely generated groups and the theory of invariant means[END_REF], which corresponds to an earlier construction by Chabauty [START_REF] Chabauty | Limite d'ensembles et géométrie des nombres[END_REF]; for introductory expositions, see [START_REF] Champetier | L'espace des groupes de type fini[END_REF][START_REF] Champetier | Limit groups as limits of free groups[END_REF][START_REF] Paulin | Sur la théorie élémentaire des groupes libres (d'après Sela)[END_REF]. Denote by B r the ball of radius r in F n (centered at the trivial element). Given normal subgroups

N = N ′ ⊳ F n , we set d(N, N ′ ) := exp -max r ∈ N : N ′ ∩ B r = N ∩ B r .
This turns G n into a compact, ultrametric, separable space. The map

G, (s 1 , . . . s n ) → G, (s 1 , . . . s n , 1 G )
defines an isometric embedding G n → G n+1 for all n ∈ N. We denote by G the direct limit of this directed system of topological spaces and call G the space of (finitely generated) marked groups. Note that G n is open and closed in G for all n. Given N ⊳ F n and N k ⊳ F n for k ∈ N, one has lim N k = N if and only if, for all g ∈ F n : (g ∈ N ⇐⇒ g ∈ N k for k sufficiently large) and (g / ∈ N ⇐⇒ g / ∈ N k for k sufficiently large).

3. From actions without fixed point to uniform actions 3.1. Being far from almost fixed points. Let (G, S) be a marked (finitely generated) group. Let it act by isometries on a metric space (X, d) and set δ(x) = max{d(x, sx) : s ∈ S} for any x ∈ X. A point x ∈ X is fixed by G if and only if δ(x) = 0, and the action almost has fixed points if and only if inf{δ(x) : x ∈ X} = 0. Remark 3.1. As G is finitely generated, a G-action on an R-tree T almost has fixed points if and only if it has a fixed point. Indeed, if the action on T has no fixed point, there exists g ∈ G which induces a hyperbolic isometry of T -this can be deduced from [Tig79, Cor 2.3]; see also [MS84, Prop II.2.15] or [Bes02, Exercise 2.8]. Thus, the G-action on T is uniform. This property is specific to R-trees, as illustrated in Examples 2.1 and 2.2. For general metric spaces, this Section will explain how to produce uniform actions from actions without fixed points. Lemma 3.2. We have |δ(x)δ(y)| ≤ 2d(x, y) for all x, y ∈ X.

Proof. For any s ∈ S, the triangle inequality gives d(x, sx) ≤ d(x, y) + d(y, sy) + d(sy, sx) ≤ d(y, sy) + 2d(x, y) , which implies δ(x) ≤ δ(y) + 2d(x, y). We then deduce similarly δ(y) ≤ δ(x) + 2d(x, y).

We now introduce one key ingredient to produce uniform actions, which has been directly inspired from Lemma 6.3 in [START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]; see also Proposition 4.1.1 in [START_REF] Korevaar | Global existence theorems for harmonic maps to non-locally compact spaces[END_REF] and (the proof of) Lemma 2 in [START_REF] Lafforgue | Une remarque sur les fonctions conditionnellement de type négatif[END_REF]. It asserts, that we may find points which are, roughly speaking, far from almost fixed points.

Lemma 3.3. Assume the space X is complete and the action has no fixed point. Then, for all n ∈ N * , there exists x n ∈ X such that:

for all y ∈ X, d(y, x n ) ≤ nδ(x n ) =⇒ δ(y) ≥ δ(x n ) 2 .
Proof. Let us assume by contradiction that there exists some n ∈ N * such that, for all x ∈ X, there exists y = y(x) ∈ X which satisfies both d(y, x) ≤ nδ(x) and δ(y) < δ(x)/2. Let us now take some z 0 ∈ X (recall that X is non-empty). Then, we define inductively a sequence of points

z k such that d(z k+1 , z k ) ≤ nδ(z k ) and δ(z k+1 ) < δ(z k )/2 for all k ∈ N. Consequently, we have δ(z k ) < δ(z 0 )/2 k , whence d(z k+1 , z k ) ≤ nδ(z 0 )/2 k .
Since X is complete, this shows that z k converges to some point z as k tends to ∞. Thanks to Lemma 3.2, we obtain δ(z) = lim k→∞ δ(z k ) = 0. Hence, z is a fixed point of the G-action, a contradiction.

Note that the hypotheses on X made in Lemma 3.3 cannot be dropped. Indeed, to obtain counterexamples, consider Z acting by rotations on C, respectively C \ {0}.

3.2.

Ultrafilters and ultralimits of metric spaces. Bourbaki defines ultrafilters to be maximal filters [START_REF] Bourbaki | Éléments de mathématique. Topologie générale[END_REF]. However, we think slightly differently to ultrafilters in this note.

Definition 3.4. An ultrafilter on some (non-empty) set E is a finitely-additive, {0, 1}-valued measure on P(E), that is a function ω : P(E) → {0, 1} which satisfies: (i) ω(A ∪ B) = ω(A) + ω(B) whenever A ∩ B = ∅, and (ii) ω(E) = 1 Note that (i) and (ii) imply ω(∅) = 0. In this note, we shall only need to consider ultrafilters on N. The following well-known Lemma establishes the equivalence with Bourbaki's definition. Its proof is given for completeness.

Lemma 3.5. A function ω : P(E) → {0, 1} is an ultrafilter if and only if F = {A ∈ P(E) : ω(A) = 1} satisfies:

(1) ∅ / ∈ F and E ∈ F ;

(2

) if A ∈ F and A ⊆ B ⊆ E, then B ∈ F ; (3) if A, B ∈ F , then A ∩ B ∈ F ; (4) for any A ⊆ E, one has A ∈ F or E \ A ∈ F .
Remark 3.6. Properties (1)-(3) are precisely the axioms of filters in [START_REF] Bourbaki | Éléments de mathématique. Topologie générale[END_REF].

Proof. Suppose first ω is an ultrafilter. Then (1), (2) and (4) are obvious. If A, B ∈ F , then (2) gives

ω(A ∪ B) = 1 and (i) implies ω((A ∪ B) \ B) = 0, ω((A ∪ B) \ A) = 0 and finally ω(A ∩ B) = ω(A ∪ B) -ω((A ∪ B) \ B) -ω((A ∪ B) \ A) = 1 .
Hence (3) is proved.

Conversly, we now assume (1)-(4). Then (ii) is obvious. Properties Example 3.7. Given any a ∈ E, there is an ultrafilter δ a defined by δ a (A) = 1 if a ∈ A and δ a (A) = 0 otherwise. Such an ultrafilter is called principal. Definition 3.8. Let ω be an ultrafilter on N and let X be a metric space. A sequence (x n ) in X is said to converge to x ∈ X relative to ω if, for any neighborhood V of x, the set {n ∈ N : x n ∈ V } has ω-measure 1.

The limit of a sequence, provided it exists, is unique, and we write lim n→ω x n = x, or lim ω x n = x. From this point of view, the interesting ultrafilters are the non-principal ones: for instance, any sequence (x n ) converges to x k , relative to the principal ultrafilter δ k . The existence of non-principal ultrafilters follows from Zorn's Lemma (see e.g [START_REF] Bourbaki | Éléments de mathématique. Topologie générale[END_REF], or Exercise I.5.48 in [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]). We shall use the following well-known result implicitly in the text. It ensures that any bounded sequence of real (or complex) numbers is ω-convergent. Proposition 3.9 ([Bou71]). If X is a compact metric space and ω is an ultrafilter on N, then any sequence (x n ) in X is ω-convergent.

We now define ultralimits of metric spaces, essentially as in [START_REF] Kleiner | Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings[END_REF] or [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF], (except that we allow to "rescale" the spaces before to take the limit, as is done in the construction of asymptotic cones, for instance). Let us consider sequences (X k , d k , * k ) k∈N of pointed metric spaces and r = (r k ) of positive numbers. Set

B r = x ∈ k∈N X k : the sequence (r k d k (x k , * k )) k is bounded .
If some group G acts by isometries on the spaces X k , its diagonal action may not stabilize B r . A necessary and sufficient condition is that the sequence (r k d k (g • * k , * k )) is bounded for any generator g of G. If this condition is fulfilled, we say that G acts diagonally on B r . For any ultrafilter ω on N, we may endow B r with the pseudo-distance d ω,r (x, y) = lim ω r k d k (x k , y k ). If G acts diagonally on B r , the diagonal action is isometric. Definition 3.10. Let ω be some non-principal ultrafilter on N. The ultralimit (relative to scaling factors (r k ) and to ω) of the sequence (X k , d k , * k ) is the pointed metric space (X ω,r , d ω,r , * ω,r ), where X ω,r is the separation of (B r , d ω,r ) and * ω,r denotes either the point ( * k ) k ∈ B r , or its image in X ω,r .

Note that if G acts diagonally on B r , the diagonal action induces an isometric action on every ultralimit X ω,r , which we call again diagonal. If the sequence (X k , d k ) k is constant and if r k → 0, one gets the notion of asymptotic cone, due to Gromov [START_REF] Gromov | Groups of polynomial growth and expanding maps[END_REF][START_REF] Gromov | Asymptotic invariants of infinite groups[END_REF], and van den Dries and Wilkie [START_REF] Van Den Dries | Gromov's theorem on groups of polynomial growth and elementary logic[END_REF].

Proposition 3.11 (Gromov [Gro03]). Let G be a finitely generated group acting by isometries on complete metric spaces (X k , d k ) for k ∈ N. If these actions have no fixed point, then there exist scaling factors r k > 0 and base points * k ∈ X k such that:

(1) the group G acts diagonally on B r ;

(2) for any non-principal ultrafilter ω on N, the diagonal action of G on X ω,r is uniform. [START_REF] Korevaar | Global existence theorems for harmonic maps to non-locally compact spaces[END_REF] used the same idea to rescale spaces, and then take a limit, to produce uniform actions from actions without fixed points on CAT(0) spaces. On the other hand their construction of limits uses properties of CAT(0) spaces.

Korevaar and Schoen

Proof. Let S be a finite generating set of G and set δ k (x) = max{d(x, sx) : s ∈ S} for all x ∈ X k . By Lemma 3.3, we obtain points * k ∈ X k for all k ∈ N such that:

for all y k ∈ X k , d(y k , * k ) ≤ kδ k ( * k ) =⇒ δ k (y k ) ≥ δ k ( * k ) 2 .
We now set r k = δ k ( * k ) -1 , which are well-defined since the actions have no fixed point. Thus, we have

r k d k ( * k , s • * k ) ≤ 1 for all k, so that (1) is satisfied.
To prove (2), we consider some non-principal ultrafilter ω on N. For any y = (y k ) ∈ B r , the sequence

(d(y k , * k )/δ k ( * k )) k is bounded. Hence, for k sufficiently large, one has d(y k , * k ) ≤ kδ k ( * k ), which implies δ k (y k ) ≥ δ k ( * k )/2. For all s ∈ S, set now A s = {k ∈ N : r k d k (y k , sy k ) ≥ 1/2}. The former argument implies k ∈ s∈S A s for k large enough, whence ω s∈S A s = 1. Since S is finite, there exists s ∈ S such that ω(A s ) = 1, which shows that d ω,r (y, sy) = lim ω r k d k (y k , sy k ) ≥ 1/2.
Let us now make Theorem 1.5 precise.

Theorem 3.12 (Gromov [START_REF] Gromov | Random walk in random groups[END_REF]). Let (G k , S k ) k∈N be a sequence of marked groups converging to (G, S) in the space G n . If each group G k acts without fixed point on a complete metric space (X k , d k ), then there exists scaling factors r k > 0 and base points * k ∈ X k such that:

(1) the free group F n acts diagonally on B r ;

(2) for any non-principal ultrafilter ω on N, the diagonal action of F n on the ultralimit X ω,r is uniform;

(3) the diagonal action factors through the epimorphism F n → G associated to (G, S).

Proof. Let N k and N be the normal subgroups of F n associated with (G k , S k ) and (G, S) respectively. The G k -actions on the spaces X k give F n -actions which are trivial on N k . Proposition 3.11 gives then scaling factors r k > 0 and points * k ∈ X k such that conditions (1) and (2) are fulfilled.

To prove (3), it suffices to see that the diagonal action is trivial on N . Let us take g ∈ N . As N k → N , we have g ∈ N k for k sufficiently large. Take now y = (y k ) ∈ B ω,r . Since g • y k = y k for k large enough, we get lim ω r k d k (y k , gy k ) = 0, whence d ω,r (y, gy) = 0. The subgroup N acts trivially on X ω,r , as desired.

Applications to fixed point properties

In this Section, we apply Theorem 1.5 (or Theorem 3.12) to obtain results about fixed point properties on groups. Another ingredient is to identify classes of metric spaces which are stable by ultralimits. Let us record two easy observations: Remark 4.1. When a group acts isometrically on a metric space, the action can be extended to the completion. Moreover, if the action is uniform, then so is the extension. 4.1. Fixed points in (affine) Hilbert spaces. We consider (affine) Hilbert spaces over R, by forgetting the complex structure and replacing the inner-product by its real part, if necessary. For such spaces, it is well-known that any isometry is an affine map. In Example 2.2, we exhibited an isometric action on a Hilbert space without fixed point, which almost has fixed points. On the other hand, Theorem 1.5 allows to unify the proof of Theorems 1.2 and 1.3, that we now recall. Korevaar and Schoen also prove the following: if Γ is a finitely generated group with Property (FH) and if X is a geodesically complete CAT(0) space with curvature bounded from below, then any nonuniform isometric action of Γ on X has a fixed point. To do this, they show that some limits (in their sense) of geodesically complete CAT(0) space with curvature bounded from below are Hilbert spaces (compare with Lemma 4.5 below).

Let us now mention a result by Mok in the same vein [START_REF] Mok | Harmonic forms with values in locally constant Hilbert bundles[END_REF]: if M is a compact riemannian manifold and if G = π 1 (M ), then G has property (T) if and only if, for any irreducible unitary representation π of G, there is no non-zero E π -valued harmonic 1-form (where E π is the locally constant Hilbert bundle on M induced from π). As this text was almost finished, we saw in the Appendix of [Kle] a "weak version of some results in [START_REF] Fisher | Almost isometric actions, property (T), and local rigidity[END_REF]", which implies Theorem 4.4. The proof in [Kle] uses ultralimits of Hilbert spaces in a very similar way as in this note (with less details).

For proofs of Theorems 4.3 and 4.4, we use the following easy observation.

Lemma 4.5. Any ultralimit of affine Hilbert spaces is an affine Hilbert space.

Proof. Let us consider a sequence (H k , d k , * k ) k∈N of (pointed) affine Hilbert spaces, scaling factors (r k ) k∈N and some non-principal ultrafilter ω on N. We denote by H 0 k the (Hilbert) vector space under H k . Let (H 0 ω,r , d ω,r , 0 ω,r ) be the ultralimit of the sequence (H 0 k , ||.|| k , 0), relative to (r k ) and to ω. Then, H 0 ω,r is a vector space with respect to operations (u k ) + (v k ) := (u k + v k ) and λ • (v k ) := (λ • v k ), and the formula u v := lim ω r k u k r k v k defines a bilinear form on H 0 ω,r . Moreover, for all u, v ∈ H 0 ω,r , we have:

d ω,r (u, v) 2 = lim k→ω d k (r k u k , r k v k ) 2 = lim k→ω r k (u k -v k ) r k (u k -v k ) = u -v u -v ,
so that H 0 ω,r is an inner-product space. Since any ultralimit of metric spaces is complete 3 , H 0 ω,r is a Hilbert space.

Finally, we consider the ultralimit (H ω,r , d ω,r , * ω,r ) of the sequence (H k , d k , * k ), relative to (r k ) and to ω. The action (u k ) + (x k ) := (u k + x k ) turns it into an affine space over H 0 ω,r . Hence it is an affine Hilbert space.

Proof of Theorems 4.3 and 4.4. Let (G k ) be a sequence in G which converges to some G ∈ G. As the subspaces G n form an open cover of G, we find n ∈ N * such that G, G k ∈ G n . Assuming that every G k admits an isometric action without fixed point on some affine Hilbert space H k , Theorem 1.5 gives then an ultralimit H = H ω,r on which G acts uniformly. Moreover, H is an affine Hilbert space by Lemma 4.5.

This proves Theorem 4.3. Moreover, if we specialize to the case G k = G 0 and H k = H 0 for all k, it also proves the non-trivial part of Theorem 4.4. 4.2. Fixed points in (complete) R-trees. Let us recall that R-trees have been invented by Tits [START_REF] Tits | Theorem of Lie-Kolchin" for trees[END_REF] 4 . Let us also recall (see e.g. Lemmata 1.2.6 and 2.4.3 in [START_REF] Chiswell | Introduction to Λ-trees[END_REF]) that a metric space (X, d) is an R-tree if and only if the following conditions both hold:

3. See e.g. Lemma 2.4.2 in [START_REF] Kleiner | Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings[END_REF] or Lemma I.5.53 in [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF] 4. Unlike the definition we follow, Tits required R-trees to be complete.

(1) it is geodesic, that is, for any x, y ∈ X, there exists a map c :

[0, ℓ] → X such that c(0) = x, c(ℓ) = y and d(c(t), c(t ′ )) = |t -t ′ | for all t, t ′ ∈ [0, ℓ];
(2) it is 0-hyperbolic, that is, for all x, y, z, t ∈ X: d(x, y) + d(z, t) ≤ max{d(x, z) + d(y, t), d(y, z) + d(x, t)} .

A map c as in point (1) is called a geodesic from x to y. In what follows, we shall need the fact that any ultralimit of R-trees is an R-tree. One may argue by saying that the ultralimit is a quotient of some subtree of a Λ-tree (where Λ is the ultrapower of R with respect to ω), in which we identify points at infinitesimal distance. However, we prefer a more pedestrian way.

Lemma 4.6.

(1) Any ultralimit of geodesic spaces is a geodesic space;

(2) Any ultralimit of 0-hyperbolic spaces is a 0-hyperbolic space.

This Lemma is easy. For instance, part (1) is an exercise in [START_REF] Bridson | Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften[END_REF]. We nevertheless give a proof for completeness.

Proof. Let us consider some non-principal ultrafilter ω, pointed metric spaces (X k , d k , * k ), and scaling factors r k , for k ∈ N.

(1) Assume the spaces X k are geodesic and take x = y ∈ X ω,r , represented by elements (x k ), (y k ) in B r . We set ℓ k = d k (x k , y k ) and consider geodesics c k : [0, ℓ k ] → X k from x k to y k (note that we may assume x k = y k for all k). Setting ℓ = lim ω r k ℓ k = d ω,r (x, y), we define a map (2) Assume the spaces X k are 0-hyperbolic and take x, y, z, t ∈ X ω,r , which are represented by elements (x k ), (y k ), (z k ), (t k ) in B r . Fixing ε > 0, we have As ε is arbitrary, this shows that the ultralimit X ω,r is 0-hyperbolic.

We now recall and prove Theorem 1.4 of the Introduction.

  (1) et (3) imply:[START_REF]In fact, use of Remark 4.8 is superfluous, as it is known[END_REF] for any C, D ⊆ E such that C ∩ D = ∅, one has C / ∈ F or D / ∈ F . To prove (i), let us now take A, B ⊆ E such that A ∩ B = ∅. The case ω(A) = 1 = ω(B) is impossible by[START_REF]In fact, use of Remark 4.8 is superfluous, as it is known[END_REF]. In case ω(A) = 1 and ω(B) = 0 (or ω(A) = 0 and ω(B) = 1), property (2) implies ω(A ∪ B) = 1. Finally, if ω(A) = 0 = ω(B), (4) gives E \ A ∈ F and E \ B ∈ F . We deduce then (E \ A) ∩ (E \ B) ∈ F by (3) and ω(A ∪ B) = 0 by[START_REF]In fact, use of Remark 4.8 is superfluous, as it is known[END_REF]. Hence (i) is proved.

  Remark 4.2. The subsets G n being an open cover of G, a property defines an open set in G if and only if it defines an open set in every G n .

  Theorem 4.3(Shalom [Sha00]). Property (FH) defines an open subset in G.Theorem 4.4 (Korevaar-Schoen [KS97], Shalom[START_REF] Shalom | Rigidity of commensurators and irreducible lattices[END_REF]). A finitely generated group G has Property (FH) if and only if every isometric G-action on a Hilbert space almost has fixed points.

c

  : [0, ℓ] → B r ; t → c k t ℓ k ℓ k∈N Then d(c(t), c(t ′ )) = lim ω r k d(c k (tℓ k /ℓ), c k (t ′ ℓ k /ℓ)) = lim ω r k (ℓ k /ℓ) • |tt ′ | = |tt ′ | for all t, t ′ ∈ [0, ℓ].Hence c is a geodesic from x to y.

  ω k ∈ N : r k d k (x k , z k ) ≤ d ω,r (x, z) + ε 2 , r k d k (y k , t k ) ≤ d ω,r (y, t) + ε 2 r k d k (y k , z k ) ≤ d ω,r (y, z) + ε 2 , r k d k (x k , t k ) ≤ d ω,r (x, t) + ε 2 = 1 .We now use 0-hyperbolicity of the spaces X k , which shows that the setk ∈ N : r k d k (x k , y k ) + r k d k (z k , t k ) ≤ max d ω,r (x, z) + d ω,r (y, t) + ε, d ω,r (y, z) + d ω,r (x, t) + εhas ω-measure 1. Hence, we obtaind ω,r (x, y) + d ω,r (z, t) ≤ max d ω,r (x, z) + d ω,r(y, t), d ω,r (y, z) + d ω,r (x, t) + ε .

Theorem 4. 7 (

 7 Culler-Morgan [CM87]). Property (FRA) defines an open subset in G.Remark 4.8. Let G be a finitely generated group. Then, G has property (FRA) if and only if every G-action on an R-tree has a fixed point.Proof. The completion of an R-tree is an R-tree[START_REF] Imrich | On metric properties of tree-like spaces[END_REF] -see also[START_REF] Morgan | Valuations, trees, and degenerations of hyperbolic structures[END_REF] Cor II.1.10] or [Chi01, Thm 2.4.14]. Hence, we are done by Remarks 3.1 and 4.1.

Proof of Theorem 4.7. Let (G k ) be a sequence in G n which converges to some point G ∈ G n , and such that any G k acts without fixed point on some complete R-tree T k . By remark 4.2, it suffices to show that G acts without fixed point on some complete R-tree.

Theorem 1.5 gives an ultralimit T = T ω,r of the R-trees T k on which G acts uniformly, and T is an R-tree by Lemma 4.6. Hence, Remark 4.8 concludes the proof. 5 Remark 4.9. The last proof does not work for simplicial trees: we used the fact that the class of R-trees is closed under ultralimits.

In fact, as the referee pointed out, using Theorem 1.5 to prove Theorem 4.7 is a little awkward: as Remark 3.1 tells us immediately that the G k -actions on the R-trees T k are uniform, it is unnecessary to make a clever choice of base points as in Proposition 3.11 and Theorem 3.12.