Finite-State Machines for Horospheres in Hyperbolic Right-Angled Coxeter Groups
Metric Geometry
2025-09-01 v2 Group Theory
Abstract
Relatively little is known about the discrete horospheres in hyperbolic groups, even in simple settings. In this paper we work with hyperbolic one-ended right-angled Coxeter groups and describe two graph structures that mimic the intrinsic metric on a classical horosphere: the Rips graph and the divergence graph (the latter due to Cohen, Goodman-Strauss, and Rieck). We develop, analyze, and implement algorithms based on finite-state machines that draw large finite portions of these graphs, and deduce various geometric corollaries about the path metrics induced by these graph structures.
Cite
@article{arxiv.2406.18774,
title = {Finite-State Machines for Horospheres in Hyperbolic Right-Angled Coxeter Groups},
author = {Noah Jillson and Daniel N. Levitin and Pramana Saldin and Katerina Stuopis and Qianruixi Wang and Kaicheng Xue},
journal= {arXiv preprint arXiv:2406.18774},
year = {2025}
}
Comments
37 pages, 8 figures. To appear in $\textit{Geometriae Dedicata}$