Groups acting purely loxodromically on products of hyperbolic graphs

TL;DR

This paper studies groups acting on products of hyperbolic graphs where all infinite order elements act loxodromically, revealing structural properties and distinguishing them from certain well-known groups like mapping class groups.

Contribution

It provides strong structure theorems for such groups acting on locally finite hyperbolic graph products, excluding some groups like mapping class groups from this class.

Findings

Mapping class groups of genus ≥ 3 are not in this subclass.

Groups with locally finite hyperbolic graph actions have specific structural properties.

Proper actions on products of quasitrees require non-locally finite graphs.

Abstract

We consider the class of countable groups possessing an action on a finite product of hyperbolic graphs where every infinite order element acts loxodromically. When the graphs are locally finite, we obtain strong structure theorems for the groups in this subclass, so that mapping class groups of genus at least 3 (and $Aut(F_n)$ and $Out(F_n)$ for $n\geq 4$) are not in this subclass. This contrasts with the general case, where Bestvina, Bromberg and Fujiwara showed the existence of proper actions of mapping class groups on a finite product of quasitrees. In particular these quasitrees cannot be locally finite.