Detecting virtual homomorphisms via Banach metrics

TL;DR

This paper introduces Banach metrics on infinite groups to detect virtual homomorphisms into integers, extending Cayley graph concepts and providing a new method via metric functional boundaries.

Contribution

It defines Banach metrics on groups and demonstrates their effectiveness in detecting virtual homomorphisms, surpassing Cayley graph limitations.

Findings

Detection via Cayley graph boundaries isn't always possible.

Banach metrics enable detection of virtual homomorphisms.

Provides a new framework for analyzing group homomorphisms.

Abstract

We introduce the notion of "Banach metrics" on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into Z, the additive group of integers. We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn't always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.