Growth of finitely generated subgroups of the topological full groups of inverse semigroups of bounded type

TL;DR

This paper proves that under certain conditions, finitely generated subgroups of topological full groups of inverse semigroups of bounded type exhibit subexponential growth, extending understanding of their algebraic properties.

Contribution

It establishes a link between the finiteness of incompressible elements and the subexponential growth of certain finitely generated subgroups.

Findings

Finitely generated subgroups have subexponential growth.

Growth bound is a bounded power in the exponent.

Results apply to orbit equivalent subgroups.

Abstract

Given an inverse semigroup $G_0$ of bounded type, we show, along with some other assumptions, that if the set of incompressible elements of $G_0$ is finite, then any finitely generated subgroup $G$ of the topological full group $\mathsf{F}(G_0)$ that is orbit equivalent to $G_0$ has subexponential growth with a bounded power in the exponent.