Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity
Sam Hughes
University of Oxford, Oxford, UKMotiejus Valiunas
Universytet Wrocławski, Wrocław, Poland
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Abstract
We construct a 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 group of flat-rank , 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.
Cite this article
Sam Hughes, Motiejus Valiunas, Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity. Comment. Math. Helv. 99 (2024), no. 2, pp. 397–436
DOI 10.4171/CMH/572