Gromov's random monsters do not act non-elementarily on hyperbolic spaces

TL;DR

This paper demonstrates that Gromov's random monster groups, constructed from random labelings of expander graphs, cannot perform non-elementary actions on hyperbolic spaces, revealing limitations in their geometric group actions.

Contribution

It establishes that Gromov's monster groups do not admit non-elementary hyperbolic actions, contrasting with previous expectations about their geometric properties.

Findings

Gromov's monster groups cannot act non-elementarily on hyperbolic spaces

Analysis of random walks links group properties to graph labelings

The proof uses properties of expander graphs and random walks

Abstract

We show that Gromov's monster groups arising from i.i.d. labelings of expander graphs do not admit non-elementary actions on geodesic hyperbolic spaces. The proof relies on comparing properties of random walks on randomly labeled graphs and on groups acting non-elementarily on hyperbolic spaces.