Stability properties of multiplicative representations of free groups

TL;DR

This paper extends the class of multiplicative representations of free groups, demonstrating their stability under various group operations, which facilitates constructing new representations for virtually free groups.

Contribution

It introduces a broader class of multiplicative representations that remains stable under sums, automorphisms, and subgroup restrictions, enhancing the understanding of free group representations.

Findings

Mult(G) is stable under finite direct sums.

Mult(G) is invariant under automorphisms of G.

Stability under restriction and induction from finite index subgroups.

Abstract

We extend the construction of multiplicative representations for a free group G introduced by Kuhn and Steger (Isr. J., (144) 2004) in such a way that the new class Mult(G) so defined is stable under taking the finite direct sum, under changes of generators (and hence is Aut(G)-invariant), under restriction to and induction from a subgroup of finite index. The main tool is the detailed study of the properties of the action of a free group on its Cayley graph with respect to a change of generators, as well as the relative properties of the action of a subgroup of finite index after the choice of a "nice" fundamental domain. These stability properties of Mult(G) are essential in the construction of a new class of representations for a virtually free group (Iozzi-Kuhn-Steger, arXiv:1112.4709v1)