Geometric and analytic aspects of infinite groups

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We study infinite groups via their actions on various classes of spaces, with a particular emphasis on two types of actions, in some sense extreme:(a) Actions with a global fixed point. The property (called fixed point property) of a group of having only such actions on spaces in a given class may have strong implications. Kazhdan's property (T) is the most important version of fixed point property. Taking finite quotients of groups with property (T) is one of the most used ways to construct families of expanders. (b) Proper actions. This means on the contrary that only finitely many elements in the group translate a point in a compact set to a point in the same set. In other words, each orbit of the group is a faithful enough picture of the group itself, drawn on the blackboard'' provided by a space in the given collection. Various versions of amenability are connected with such actions.We focus on actions on the following classes of spaces:(1) Hilbert spaces and Banach spaces. Hilbert spaces, which are in some sense infinite dimensional generalisations of the familiar Euclidean spaces, seem the ideal blackboard'' on which to draw an infinite group. Surprisingly enough, a proper embedding of an infinite group in a Hilbert space (more generally in a uniformly convex Banach space) is not granted, its very existence, as well as the parameter called compression measuring how much this embedding distorts the group, encapsulate a lot of information on the group. The Rapid Decay property, an important information on the C-star algebra of the group, relevant to the Novikov and Baum-Connes conjectures via Vincent Lafforgue's work, is also defined in terms of an action (linear this time) of the group on the Hilbert space of square-summable real functions on it.(2) CAT(0) spaces (i.e. non-positively curved spaces, in a metrical sense). Interesting particular cases are the cube complexes (with one-skeleta the median graphs) and their non-discrete generalisations the median spaces, and real trees.(3) Symmetric spaces. The most important actions on such spaces are those ofarithmetic lattices (such as the group of square matrices with integer entries); they have close connections with various Number Theory problems. The understanding of such actions brings valuable information on the geometry of arithmetic lattices, some of the most interesting infinite groups.(4) Actions on limit spaces, appearing as limit actions of groups, in problems of compactification of spaces of representations. These actions relate to several interesting topics mixing group theory and logic: they are used in the recent solution of the Tarski conjecture; the possible number of different limit spaces for a group also relates to the Continuum Hypothesis.