Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

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.

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!

References in corpus (5)

• 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
• Helly groups, coarsely Helly groups, and relative hyperbolicity
• Characterizing Biautomatic Groups

Cited by in corpus (2)

• Model geometries of finitely generated groups
• Higher-rank GBS groups: non-positive curvature and biautomaticity