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.)[-]

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.)[-]

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 $II_1$.[-]

Kazhdan projections are usually considred objects relevant in operator algebras. In particular, they played a central part in the construction of counter-examples to the Baum-Connes conjecture.
In this talk I shall explain how, in the general setting of a family of representations on Banach spaces, one can reformulate the Kazhdan property "almost invariant implies invariant vectors" in terms of Kazhdan projections, providing also an explicit formula of the latter, using Markov operators associated to a random walk on the group. I will then explain some applications of this new approach.
This is joint work with Piotr Nowak.[-]

Given a nontrivial conjugacy class $g$ in a free group $F_{N}$, what can we say about the typical growth of g under application of a random product of auto-morphisms of $F_{N}$? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the surface. This is partly joint work with François Dahmani.[-]

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right ) $ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.[-]

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.[-]