Rigid topologies on groups

TL;DR

This paper proves that every infinite, countable, residually finite group can be endowed with a Hausdorff topology that is neither discrete nor precompact, expanding understanding of possible topological structures on such groups.

Contribution

It introduces a new class of topologies on residually finite groups, demonstrating the existence of non-trivial, non-precompact Hausdorff group topologies.

Findings

Existence of non-discrete, non-precompact Hausdorff topologies on certain groups

Extension of topological group theory to residually finite groups

New insights into the structure of topologies on infinite groups

Abstract

Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.