On Infinite Groups and Their Actions: From Group Actions On Graphs To Group Actions On Complexes

Understanding how a group can act on a given type of space can be a valuable tool for proving properties of both the group and the space. This thesis focuses on three distinct topics involving actions of infinite groups on graphs, cube complexes and metric spaces.

In the first part, we answer a question by Grigorchuk asking whether it is possible to give an explicit and elementary description of a CAT(0) cube complex on which the groups he defined act and, if so, to describe these ac- tions. Recall that the Grigorchuk groups are subgroups of finite types of the group of automorphisms of the rooted binary tree. This family of groups has been a prolific subject of study and has provided answers to many problems. The main ingredient allowing us to build such a complex is the existence of a Schreier graph with two ends which come from the action of these groups on the boundary of the binary tree. Using this graph, it is possible to construct a CAT(0) cube complex on which all Grigorchuk groups act without bounded orbit, or, equivalently, without a fixed point. Moreover, for a non-countable subfamily of Grigorchuk groups, we show that this action is faithful and proper. In this case, the CAT(0) cube complex is a model for the classifying space of the proper actions of the group.

In the following, we present the results of joint projects with P-H. Leemann where we are interested in the stability of certain properties of groups for the wreath product. We start by giving a proof of a result about the stability of the proprety FW for the wreath product which is very close to a theorem of Cherix-Martin-Valette and Neuhauser about the stability of the property (T). For this purpose, we use an obstruction to property FW described in terms of the number of ends in the Schreier graphs associated to the group.

Then, we study the stability of these properties for the wreath product in a more general framework. Recall that, for a countable group, property (T) is equivalent to the property FH defined as “every action on a Hilbert space has a fixed point”. By generalizing this example, we notice that there exists a whole family of algebraic properties associated to a group which are defined as “every action of this group on a certain type of metric space has a fixed point”. We can, for example, think about the property FA (any action on a tree has a fixed point) or the property FW (any action on a wall space has a fixed point). We show how the stability of a large family of such properties for the wreath product can be proved in a unified way.

Finally, we study the notion of expansion for objects of dimension greater than 1. We start by defining the notion of boundary expansion for CW com- plexes and then prove a link between this expansion and the spectrum of the Laplacian, in the same spirit as the Cheeger-Buser inequalities for graphs.