On the commensurability of hyperbolic Coxeter groups

TL;DR

This paper investigates the relationships between hyperbolic Coxeter groups of finite covolume, establishing necessary conditions for their commensurability and exploring the algebraic fields and generators associated with these groups.

Contribution

It introduces three necessary conditions for commensurability and new insights into the fields of definition and generators of hyperbolic Coxeter groups.

Findings

Three necessary conditions for commensurability.

Characterization of possible fields of definition.

Two new sets of generators for quasi-arithmetic groups.

Abstract

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we tackle different topics around the field of definition of a hyperbolic Coxeter group: which possible fields can arise, how this field determines a range of possible dihedral angles of a Coxeter polyhedron and we provide two new sets of generators for quasi-arithmetic groups. This work is a concise version of chapters 4 and 5 of the author's Ph.D. thesis.