Random extensions of free groups and surface groups are hyperbolic

TL;DR

This paper proves that random extensions of free groups and surface groups are hyperbolic, with implications for the structure of random subgroups in hyperbolic and weakly hyperbolic groups.

Contribution

It establishes that random extensions generated by independent random walks are hyperbolic, extending understanding of the geometric properties of such algebraic constructions.

Findings

Random extensions of free groups are hyperbolic.

Random extensions of surface groups are hyperbolic.

A random subgroup of a weakly hyperbolic group is free and undistorted.

Abstract

In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out$(F_N)$ or Mod$(S_g)$ generated by $k$ independent random walks. Our main theorem has several applications, including that a random subgroup of a weakly hyperbolic group is free and undistorted.