On the commensurability of hyperbolic Coxeter 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.