Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity

Abstract

We construct a $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ 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.