Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity
arXiv:2203.11996 · doi:10.4171/cmh/572
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.
arXiv commentv3: 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!
References in corpus (10)
- Isometry groups of non-positively curved spaces: discrete subgroups
- Geometry of the mapping class group II: A biautomatic structure
- Commensurating HNN-extensions: non-positive curvature and biautomaticity
- Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness
- Graphs and complexes of lattices
- Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups
- Hierarchically hyperbolic groups and uniform exponential growth
- Helly groups, coarsely Helly groups, and relative hyperbolicity
- 2-dimensional Coxeter groups are biautomatic
- Characterizing Biautomatic Groups