On the dimension growth of groups

TL;DR

This paper investigates the dimension growth of groups, showing that solvable subgroups of Thompson's group F have polynomial growth, while F itself and some solvable groups of class 3 exhibit exponential growth, linking these properties to graph expansion and Ramsey theory.

Contribution

It establishes the polynomial and exponential bounds on the dimension growth of specific groups and explores their connections to graph expansion and Ramsey theory.

Findings

Solvable finitely generated subgroups of F have polynomial dimension growth.

The group F and some solvable groups of class 3 have exponential dimension growth.

Connections between dimension growth, graph expansion, and Ramsey theory are described.

Abstract

Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections between dimension growth, expansion properties of finite graphs and the Ramsey theory.