Related papers: Contracting elements and random walks

Stallings remarked that an outer automorphism of a free group may be thought of as a subdivision of a graph followed by a sequence of folds. In this thesis, we prove that automorphisms of fundamental groups of graphs of groups satisfying…

Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group,…

The question which motivates the article is the following: given a group acting on a CAT(0) cube complex, how can we prove that it is acylindrically hyperbolic? Keeping this goal in mind, we show a weak acylindricity of the action on the…

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main…

Given a geodesic metric space $X$, we construct a corresponding hyperbolic space, which we call the contraction space, that detects all strongly contracting directions in the following sense; a geodesic in $X$ is strongly contracting if and…

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices…

We prove that if $\phi,\psi\in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<\phi,\psi>\le Out(F_N)$ is not virtually cyclic then some high powers of $\phi$ and $\psi$ generate a free subgroup of rank two,…

We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These…

We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least…

For any proper action of a non-elementary group $G$ on a proper geodesic metric space, we show that if $G$ contains a contracting element, then there exists a sequence of proper quotient groups whose growth rate tends to the growth rate of…

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every…

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists…

We prove a rigidity result for group actions on the line whose elements have what we call "hyperbolic-like" dynamics. Using this, we give a spectral rigidity theorem for $\mathbb{R}$-covered Anosov flows on 3-manifolds, characterizing orbit…

Continuing from the author's previous article 'Random walks and contracting elements I', we study random walks on (possibly asymmetric) metric spaces using the bounded geodesic image property (BGIP) of certain isometries. As an application,…

In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…

The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization…

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of…

The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary…

Let $S$ be a rank-one symmetric space of non-compact type and let $X$ be a $\text{CAT}(-1)$ space. A well-known result by Bourdon states that if a topological embedding $\varphi: \partial_\infty S \rightarrow \partial_\infty X$ respects…

Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$,…