Geometric and analytic aspects of infinite groups

我们研究无限群通过他们的行动对各种类别的空间，特别强调两种类型的行动，在某种意义上极端：（a）行动与全球不动点。一个群的性质（称为不动点性质）在一个给定类的空间上只有这样的作用可能有很强的含义。Kazhdan性质（T）是不动点性质的最重要形式。取具有性质（T）的群的有限直积是构造扩张子族最常用的方法之一。(b)适当的行动。相反，这意味着群中只有1000个元素将紧集合中的一个点平移到同一集合中的一个点。换句话说，群的每个轨道都是群本身的一幅足够忠实的图画，画在给定集合中的一个空格所提供的黑板上。各种形式的顺从性都与这种作用有关，我们主要讨论以下几类空间上的作用：（1）Hilbert空间和Banach空间。希尔伯特空间在某种意义上是我们熟悉的欧几里得空间的无限维推广，它似乎是画无限群的理想黑板。令人惊讶的是，一个无限群在希尔伯特空间（更一般地说是在一致凸的Banach空间）中的适当嵌入是不被允许的，它的存在，以及被称为压缩的参数，测量这种嵌入对群的扭曲程度，包含了很多关于群的信息。快速衰减属性，一个重要的信息，对C-星代数的小组，有关诺维科夫和鲍姆-康纳斯代数通过文森特Lafforgue的工作，也被定义在行动（线性的这一次）的希尔伯特空间的平方求和的真实的功能的小组。(2)CAT（0）空间（即度量意义上的非正曲空间）。有趣的特殊情况是立方体复合体（与一个-λ的中位数图）和他们的非离散推广中位数空间，和真实的树。(3)对称空间。在这样的空间上最重要的作用是算术格的作用（如整数元方阵群）;它们与各种数论问题有着密切的联系。对这种行为的理解为算术格的几何学带来了有价值的信息，算术格是一些最有趣的无限群。(4)作用于极限空间，表现为群的极限作用，在表示空间的紧化问题中。这些行动涉及到几个有趣的主题混合群论和逻辑：他们被用于最近的解决方案的塔斯基猜想;可能的数量不同的极限空间的一组也涉及到连续统假设。

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.