[Paper Review] Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

Sam Hughes, Motiejus Valiunas|arXiv (Cornell University)|Mar 22, 2022

Geometric and Algebraic Topology45 references4 citations

TL;DR

This paper constructs the first example of a hierarchically hyperbolic group (HHG) that is not biautomatic, using a commensurating HNN extension of an arithmetic surface group acting on $\mathbb{RH}^2 \times \mathcal{T}_{24}$. The group is $\mathrm{CAT}(0)$, torsion-free, uniform, and irreducible, with non-biautomaticity proven via geodesic currents and failure of rationality in translation length functions under a hypothetical biautomatic structure.

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.

Motivation & Objective

• To resolve the open question of whether all hierarchically hyperbolic groups (HHGs) are biautomatic.
• To construct the first example of a $\mathrm{CAT}(0)$ group of flat-rank 2 that is not biautomatic.
• To provide the first example of an HHG that is injective but not Helly, and a non-residually finite HHG.
• To develop a novel method for proving non-biautomaticity based on geodesic currents, independent of boundary analysis.
• To show that the failure of biautomaticity in HHGs cannot be reduced to known constructions like Leary–Minasyan groups.

Proposed method

• Construct a uniform irreducible lattice $\Gamma < \mathrm{PSL}_2(\mathbb{R}) \times T_{24}$ via an HNN extension of an arithmetic surface group, where the stable letter commensurates the surface group and acts as an infinite-order elliptic isometry on $\mathbb{RH}^2$.
• Use the isometric action of $\Gamma$ on $\mathbb{RH}^2 \times \mathcal{T}_{24}$ to deduce that $\Gamma$ is $\mathrm{CAT}(0)$ and acts freely cocompactly.
• Apply [Hug22a, Corollary 3.3] to conclude $\Gamma$ is a hierarchically hyperbolic group (HHG).
• Assume a uniformly finite-to-one biautomatic structure $ (B, \mathcal{M}) $ on $\Gamma$, and derive a biautomatic structure $ (A, \mathcal{L}) $ on the surface subgroup $\widehat{G} \cong \pi_1(\Sigma)$ via quasiconvexity and restriction.
• Define the translation length function $\tau_{\mathcal{L}}$ on $\mathcal{C}^+(\Sigma)$, which factors through the geodesic current space $\mathcal{G}^+(\Sigma)$ and is shown to be positively linear and homogeneous.
• Use deep results from [Martínez-Granado and Thurston] on extending functions to geodesic currents to show $\tau_{\mathcal{L}} = k \cdot \iota(-, \lambda_\Sigma)$, where $\iota$ is the intersection number, leading to a contradiction via irrationality of $\tau_{\mathcal{L}}(\gamma_a)/\tau_{\mathcal{L}}(\gamma_c)$.

Experimental results

Research questions

• RQ1 Is every hierarchically hyperbolic group (HHG) biautomatic?
• RQ2 Can a $\mathrm{CAT}(0)$ group of flat-rank 2 fail to be biautomatic?
• RQ3 Does there exist a non-biautomatic HHG that is not constructed from or contains a Leary–Minasyan group?
• RQ4 Can the failure of biautomaticity be detected via geodesic currents and intersection numbers?
• RQ5 Is there a $\mathrm{CAT}(0)$ group that is injective but not Helly, or non-residually finite, and still HHG?

Key findings

• The paper constructs the first known example of a hierarchically hyperbolic group (HHG) that is not biautomatic, answering an open question in the field.
• The constructed group $\Gamma$ is a uniform irreducible lattice in $\mathrm{PSL}_2(\mathbb{R}) \times T_{24}$, acting freely cocompactly on $\mathbb{RH}^2 \times \mathcal{T}_{24}$, and is therefore $\mathrm{CAT}(0)$.
• The group $\Gamma$ has flat-rank 2, providing the first example of a $\mathrm{CAT}(0)$ group of flat-rank 2 that is not biautomatic.
• The non-biautomaticity is established via a contradiction: a hypothetical biautomatic structure leads to a translation length function $\tau_{\mathcal{L}}$ that must be a rational multiple of the intersection number $\iota(-, \lambda_\Sigma)$, but the ratio $\tau_{\mathcal{L}}(\gamma_a)/\tau_{\mathcal{L}}(\gamma_c)$ is irrational.
• The group $\Gamma$ is injective but not Helly, providing the first example of an HHG with this property.
• The result implies that at least one of the following is false: (1) every HHG is automatic, or (2) every automatic group is biautomatic.