Exponentially generic subsets of groups

TL;DR

This paper investigates the typical properties of subgroups in hyperbolic groups and the complexity of the word problem in amenable groups, revealing that random elements often generate free subgroups while some amenable groups have unsolvable word problems on large subsets.

Contribution

It demonstrates that generic subgroups in hyperbolic groups are free and embedded, and identifies amenable groups with universally unsolvable word problems on large subsets.

Findings

Random sets in hyperbolic groups are likely to generate free subgroups.

Some amenable groups have unsolvable word problems on all large subsets.

Generic behavior in these groups differs significantly from special cases.

Abstract

In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words.