Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity (2203.11996v3)

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.

• The paper constructs the first known example of a hierarchically hyperbolic group that fails to be biautomatic, addressing a prominent question in geometric group theory.
• It establishes instances of non-biautomatic CAT(0) groups of flat-rank 2, challenging existing theories about CAT(0) group properties.
• The research demonstrates a case where a group is injective but not Helly, highlighting a key distinction since Helly groups are known to be biautomatic.

Commensurating HNN-Extensions: Hierarchical Hyperbolicity and Biautomaticity

This paper addresses several intriguing problems at the interface of geometric group theory, $CAT(0)$ spaces, hierarchical hyperbolicity, and automaticity of groups. The authors construct the first known example of a hierarchically hyperbolic group (HHG) that fails to be biautomatic. Additionally, they establish instances of non-biautomatic $CAT(0)$ groups of flat-rank 2 and demonstrate a $CAT(0)$ group being injective but not Helly. The overarching goal is to explore the conditions under which hierarchical hyperbolicity and automaticity interact, a question with deep implications in both theoretical and applied group theory.

Core Contributions and Methodology

1. Construction of HHG without Biautomaticity: The authors construct a $\mathrm{CAT}(0)$ hierarchically hyperbolic group (HHG) that acts geometrically on the product of a hyperbolic plane and a locally-finite tree, yet is not biautomatic. This endeavor answers a prominent question in the field since known HHGs often exhibit properties implying biautomaticity.
2. Non-Biautomatic $CAT(0)$ Group Example: This work defines a $CAT(0)$ group of flat-rank 2 which is not biautomatic, addressing the problem posed by existing theories suggesting that $CAT(0)$ groups possessing certain dimensional properties might be inherently biautomatic.
3. Discrepancy between Injective and Helly Properties: The research illustrates a case where a group is injective according to metric space definitions but fails to be Helly. This distinction is pivotal given that Helly groups are known to be biautomatic, highlighting a bifurcation in properties hitherto thought closely related.

Implications in Group Theory

The implications of this work are far-reaching, especially the conceptual separation it shows between hierarchical hyperbolicity and biautomaticity. The introduction of dense submonoids in $PSL_2(R)$ further provides a robust framework to analyze the impact of discrete geometry on group properties. Findings here lay the groundwork for reevaluating the automaticity of groups from a new structural vantage, offering the potential for novel insights into longstanding open questions, such as whether all $S$-arithmetic lattices are biautomatic.

Theoretical and Practical Future Directions

• Theoretical Exploration: Future research could explore whether other instances of groups previously thought to fit within the domain of biautomatic groups also defy this property when cast under the lens of hierarchical hyperbolicity.
• Connections with Large Scale Geometry: Given the intersection numbers and the interplay between geodesic currents, further investigation is warranted to assess how large-scale geometric properties impact intricate group-theoretic dynamics, particularly in non-positively curved spaces.
• Implications for Computational Group Theory: Practically, understanding groups that are hierarchically hyperbolic but not biautomatic may open the door to new algorithms in computational group theory, potentially leading to new methods for group isomorphism problems, and automatic structures' computations.

Overall, the paper is a significant contribution to the field, offering detailed constructions and examples that challenge existing boundaries between seemingly related properties of mathematical structures in group theory and beyond. The findings invite deeper questioning and exploration into the nature of biautomaticity in relation to other group properties, suggesting rich veins for future inquiry.