Contracting elements and random walks

TL;DR

This paper introduces a new notion of contracting elements in groups, shows their equivalence with known hyperbolic elements in various groups, and analyzes the behavior of random walks relative to these elements.

Contribution

It defines contracting and weakly contracting elements, establishes their equivalence with classical hyperbolic elements in multiple contexts, and studies their probabilistic properties in random walks.

Findings

Contracting elements coincide with hyperbolic elements in various groups.

Random walks in groups with contracting elements tend to leave these elements with exponentially small probability.

Contracting elements are contained in hyperbolically embedded elementary subgroups.

Abstract

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in $Out(F_n)$, $n\geq 3$. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability. Also, we show that each (weakly) contracting element is contained in a hyperbolically embedded elementary subgroup.