Random walks is a topic at the crossroads of probability, ergodic theory, potential theory, harmonic analysis, geometry, and graph theory. Its roots can be traced back to the famous article by Pólya in 1921, which characterizes the recurrence of random walks on $\mathbb{Z}^{d}$ in terms of the dimension $d$. When random walks take place on a group, or more generally on a homogeneous space, it provides an even richer framework for study. From a probabilistic point of view, this additional structure serves as an extra tool, facilitating the study of the behaviour of the random walk on the underlying space. Regarding groups and their actions, random walks offer a means to explore generic or non-generic parts of groups and, at times, even to demonstrate intrinsic geometric properties, as is clearly shown by Kesten's amenability criterion (1959). This is an introductory course on the topic. Emphasis will be given on the interplay between probability and the structure of the group. The course will also provide insights into current research questions. Here is an outline of each session :
(1) Equivalent of Pólya's criterion for random walks on groups and rigidity theorems : does walking randomly on a given group in two different ways affect the recurrence of the walks ?
(2) Kesten's probabilistic criterion of the amenability of a finitely generated group ; defined in this course in terms of isoperimetric profile. The tools in 1) and 2) are essentially coming from analysis on groups.
(3) Tools coming from subadditivity to study the behaviour of a random walk on a group (drift, entropy and expansion of the random walk, etc.)[-]

I will present results of three studies, performed in collaboration with M.Benli, L.Bowen, A.Dudko, R.Kravchenko and T.Nagnibeda, concerning the invariant and characteristic random subgroups in some groups of geometric origin, including hyperbolic groups, mapping class groups, groups of intermediate growth and branch groups. The role of totally non free actions will be emphasized. This will be used to explain why branch groups have infinitely many factor representations of type $II_1$.[-]

Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a PD(3) pair.
This is joint work with Genevieve Walsh.[-]

Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich-Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. Along the way, I will illustrate how our methods distinguish free-by-cyclic groups which are the fundamental group of a 3-manifold. This is joint work with Yael Algom-Kfir and Arnaud Hilion.[-]

The study of groups often sheds light on problems in various areas of mathematics. Whether playing the role of certain invariants in topology, or encoding symmetries in geometry, groups help us understand many mathematical objects in greater depth. In coarse geometry, one can use groups to construct examples or counterexamples with interesting or surprising properties. In this talk, we will introduce one such metric object arising from finite quotients of finitely generated groups, and survey some of its useful properties and associated constructions.[-]

(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f $ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.[-]

Using a very recent result of Calderon and Salter, we relate small type Artin groups defined by Coxeter diagram which are trees to mapping class groups. This gives information on both the Artin groups with respect to commensurability and hyperbolicity of the parabolic subgroup graph as well as information on the mapping class group and its associated geometric spaces, namely generating sets of finite index subgroups and fundamental groups of strata of abelian differentials. I'll try to highlight the many ways in which this reflects various aspects of Mladen's work.[-]