Convergent sequences in discrete groups

TL;DR

This paper characterizes finitely generated groups that have sequences of non-trivial elements converging to the identity in all compact images, revealing a deep connection between such sequences and the group's algebraic structure.

Contribution

It establishes a precise criterion linking the existence of convergent sequences to the group's virtual abelianness, providing a new perspective on group convergence properties.

Findings

Finitely generated groups with such sequences are not virtually abelian.

Virtually abelian groups do not admit such convergent sequences.

The result characterizes the convergence behavior in terms of algebraic structure.

Abstract

We prove that a finitely generated group contains a sequence of non-trivial elements which converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian.