On topologizable and non-topologizable groups

TL;DR

This paper introduces the first examples of hereditarily non-topologizable groups, explores their implications for c-compactness versus compactness, and proposes methods for constructing topologizable groups, answering several open questions in the field.

Contribution

It constructs the first infinite hereditarily non-topologizable groups and develops a new method for creating topologizable groups based on generic properties.

Findings

Hereditarily non-topologizable groups exist infinitely.

C-compactness does not imply compactness in topological groups.

Existence of non-discrete quasi-cyclic groups of finite exponent.

Abstract

A group $G$ is called hereditarily non-topologizable if, for every $H\le G$, no quotient of $H$ admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked $k$-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.