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September 10, 2025Open Access

Cone types and spectral radius of hyperbolic triangle groups and hyperbolic tessellations

MHMegan HowarthUniversity of Geneva TNTatiana NagnibedaUniversity of Geneva

Key Points

The number of cone types in hyperbolic triangle groups depends solely on group parameters.
Estimates for the spectral radius of random walks on tessellations are provided, both upper and lower.
Tessellations of the hyperbolic plane are characterized through their cone type structures.
The study enhances understanding of the relationship between graph properties and random walks in hyperbolic settings.

Abstract

This paper is devoted to the study of tessellations of the hyperbolic plane, especially the ones associated to hyperbolic triangle groups (l, m, n). We give a full description of the cone types of these graphs and show that their number depends only on the defining parameters of the group. We then use the cone type structure to provide estimates of the spectral radius for the simple random walk on these tessellations, from above and from below.

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