Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity
Group Theory
2024-05-14 v3 Geometric Topology
Abstract
We construct a CAT(0) 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 CAT(0) group of flat-rank 2, and the first example of an HHG which is injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.
Keywords
Cite
@article{arxiv.2203.11996,
title = {Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity},
author = {Sam Hughes and Motiejus Valiunas},
journal= {arXiv preprint arXiv:2203.11996},
year = {2024}
}
Comments
v3: Updated with referees' comments, to appear in Commentarii Mathematici Helvetici. 32 pages, 4 figures. v2: the statement of Corollary D has been strengthened. v1: Comments are welcome!