Most rigid representation and Cayley index of finitely generated groups

TL;DR

This paper investigates the Cayley index of finitely generated groups, establishing that it is 1 for some groups and 2 for others, with the latter case achieved by a finite generating set.

Contribution

It completes the classification of finitely generated groups by their Cayley index, showing the index is either 1 or 2, and proves the index 2 case is realized with a finite generating set.

Findings

Cayley index is 1 for certain groups

Cayley index is 2 for remaining groups

Index 2 is attained with a finite generating set

Abstract

If $G$ is a group and $S$ a generating set, $G$ canonically embeds into the automorphism group of its Cayley graph and it is natural to try to minimize, over all generating sets, the index of this inclusion. This infimum is called the Cayley index of the group. In a recent series of works, we have characterized the infinite finitely generated groups with Cayley index $1$. We complement this characterization by showing that the Cayley index is $2$ in the remaining cases and is attained for a finite generating set.