Growth of actions of solvable groups

TL;DR

This paper investigates the growth properties of faithful actions of finitely generated solvable groups, establishing sharp lower bounds on their Schreier growth gaps based on group structure and properties.

Contribution

It provides new bounds on Schreier growth gaps for various classes of solvable groups, including metabelian, linear, and torsion-free groups, extending understanding of their geometric action properties.

Findings

Metabelian groups have a Schreier growth gap of n^2 if finitely presented or torsion-free and not virtually abelian.

Metabelian groups of Krull dimension k have a Schreier growth gap of n^k.

Solvable groups of finite Pr"ufer rank have a Schreier growth gap of exponential growth, unless virtually nilpotent.

Abstract

Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$ is not virtually abelian. We also prove that if $G$ is a metabelian group of Krull dimension $k$, then $G$ has a Schreier growth gap $n^k$. For instance the wreath product $C_p \wr \mathbb{Z}^d$ has a Schreier growth gap $n^d$, and $\mathbb{Z} \wr \mathbb{Z}^d$ has a Schreier growth gap $n^{d+1}$. These lower bounds are sharp. For solvable groups of finite Pr\"ufer rank, we establish a Schreier growth gap $\exp(n)$, provided $G$ is not virtually nilpotent. This covers all solvable groups that are linear over $\mathbb{Q}$. Finally for a vast class of torsion-free solvable groups, which includes solvable groups that are linear, we establish a Schreier growth gap $n^2$.