On commensurable hyperbolic Coxeter groups

Abstract

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space $$\mathbb H^n$$ , new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions.