Groups with positive rank gradient and their actions

TL;DR

This paper investigates the properties of finitely generated LERF groups with positive rank gradient, demonstrating how subgroups of infinite index behave and constructing specific group actions, extending some results to profinite groups.

Contribution

It generalizes Olshanskii's theorem to a broader class of groups and constructs new group actions with particular orbit properties.

Findings

Finite product of infinite index subgroups does not cover the group

Existence of a transitive virtually faithful action with finite orbits for finitely generated subgroups of infinite index

Some results extend to profinite groups with positive rank gradient

Abstract

We show that given a finitely generated LERF group $G$ with positive rank gradient, and finitely generated subgroups $A,B \leq G$ of infinite index, one can find a finite index subgroup $B_0$ of $B$ such that $[G : \langle A \cup B_0 \rangle] = \infty$. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover $G$. We construct a transitive virtually faithful action of $G$ such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.