On the cubical geometry of Higman’s group

Abstract

We investigate the cocompact action of Higman’s group on a $CAT\left(0\right)$ square complex associated to its standard presentation. We show that this action is in a sense intrinsic, which allows for the use of geometric techniques to study the endomorphisms of the group, and we show striking similarities with mapping class groups of hyperbolic surfaces, outer automorphism groups of free groups, and linear groups over the integers. We compute explicitly the automorphism group and outer automorphism group of Higman’s group and show that the group is both Hopfian and co-Hopfian. We actually prove a stronger rigidity result about the endomorphisms of Higman’s group: every nontrivial morphism from the group to itself is an automorphism. We also study the geometry of the action and prove a surprising result: although the $CAT\left(0\right)$ square complex acted upon contains uncountably many flats, the Higman group does not contain subgroups isomorphic to ${\mathbb{Z}}^2$. Finally, we show that this action possesses features reminiscent of negative curvature, which we use to prove a refined version of the Tits alternative for Higman’s group.