Rank gradient and torsion groups

TL;DR

This paper constructs the first examples of infinite finitely generated residually finite torsion groups with positive rank gradient, demonstrating their non-amenability and exploring implications for cost and $L^2$-Betti numbers.

Contribution

It introduces the first known examples of such torsion groups with positive rank gradient, advancing understanding of their properties and applications.

Findings

Existence of infinite finitely generated residually finite torsion groups with positive rank gradient

These groups are non-amenable

Applications to cost and $L^2$-Betti numbers

Abstract

We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and $L^2$-Betti numbers are discussed.