Finite-State Machines for Horospheres in Hyperbolic Right-Angled Coxeter Groups

TL;DR

This paper develops finite-state machine algorithms to analyze graph structures modeling horospheres in hyperbolic right-angled Coxeter groups, revealing geometric properties of their path metrics.

Contribution

It introduces novel finite-state machine algorithms for graph structures mimicking horospheres in hyperbolic Coxeter groups, with analysis and implementation.

Findings

Algorithms successfully generate large graph portions

Derived geometric properties of induced path metrics

Enhanced understanding of horosphere structures in hyperbolic groups

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.