Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

Abstract

We construct a CATp0q hierarchically hyperbolic group (HHG) acting geometrically on the product of a hyperbolic plane and a locally-finite tree which is not biautomatic. This gives the first example of an HHG which is not biautomatic, the first example of a non-biautomatic CATp0q group of flat-rank 2, and the first example of an HHG which is coarsely injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.

Lattices in a product of trees, hierarchically hyperbolic groups and virtual torsion‐freeness

“…Moreover, by Corollary 3.3 Λ is hierarchically hyperbolic. Finally, note that the methods in [17] have been used in [19] to construct an HHG which is not biautomatic.…”

Model geometries of finitely generated groups

“…Groups acting geometrically on H 2 × T include arithmetic lattices in Isom(H 2 ) × PSL(Q p ), as well as more exotic non-residually finite examples constructed by Hughes-Valiunas [HV22]. Generalised Baumslag-Solitar groups of rank two were classified up to quasi-isometry by Mosher-Sageev-Whyte, Farb-Mosher and Whyte [FM00, Why01, MSW03, Why10].…”

A note on the rational homological dimension of lattices in positive characteristic

“…The author would like to thank his PhD supervisor Professor Ian Leary for his guidance, support, and suggesting of the question. This note contains material from the author's PhD thesis [Hug21a] and was originally part of [Hug21b], but was split off into a number of companion papers [Hug21c; Hug22] (see also [HV21]) at the request of the referee. This work was supported by the Engineering and Physical Sciences Research Council grant number 2127970.…”