Hyperbolic manifolds and tessellations of type {3,5,3} associated with L_2(q)

TL;DR

This paper classifies certain normal subgroups of a tetrahedral group linked to hyperbolic 3-manifolds, analyzing their symmetry groups and how finite simple groups L_2(q) act on associated tessellations.

Contribution

It provides a detailed classification of normal subgroups of the tetrahedral group related to hyperbolic tessellations and analyzes their symmetry groups and actions.

Findings

Classification of normal subgroups K with Delta/K ≅ L_2(q)

Determination of normalisers N(K) in isometry group of H^3

Analysis of L_2(q) action on hyperbolic tessellations

Abstract

We classify the normal subgroups K of the tetrahedral group Delta=[3,5,3]^+, the even subgroup of the Coxeter group Gamma=[3,5,3], with Delta/K isomorphic to a finite simple group L_2(q). We determine their normalisers N(K) in the isometry group of hyperbolic 3-space H^3, the isometry groups N(K)/K of the associated hyperbolic 3-manifolds H^3/K, and the symmetry groups N_{Gamma}(K)/K of the icosahedral tessellations of these manifolds, giving a detailed analysis of how L_2(q) acts on these tessellations.