Invariable generation does not pass to finite index subgroups

TL;DR

This paper demonstrates that the property of invariable generation in groups does not necessarily pass to finite index subgroups and that finitely generated invariably generated groups are not always finitely invariably generated, using small cancellation methods.

Contribution

It provides new counterexamples and answers to longstanding questions about the inheritance of invariable generation properties in group theory.

Findings

Invariable generation does not pass to finite index subgroups.

Finitely generated invariably generated groups are not necessarily finitely invariably generated.

Independent results by Minasyan confirm these findings.

Abstract

Using small cancellation methods, we show that the property invariable generation does not pass to finite index subgroups, answering questions of Wiegold and Kantor-Lubotzky-Shalev. We further show that a finitely generated group that is invariably generated is not necessarily finitely invariably generated, answering a question of Cox. The same results were also obtained independently by Minasyan.