On the cubical geometry of the Higman group

Alexandre Martin (University of Vienna)

Wednesday 1st June, 2016 16:00-17:00 Maths 522


The Higman group was constructed as the first example of a finitely presented infinite group without non-trivial finite quotients. Despite this exotic behaviour, I will describe striking similarities with mapping class groups of hyperbolic surfaces, outer automorphisms of free groups and special linear groups over the integers. The main object of study will be the cocompact action of the group on a CAT(0) square complex naturally associated to its standard presentation. This action, which turns out to be intrinsic, can be used to explicitly compute the automorphism group and outer automorphism group of the Higman group, and to show that the group is both Hopfian and co-Hopfian. A surprisingly stronger result actually holds: Every non-trivial morphism from the Higman group to itself is an automorphism.

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