Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a certain geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examples conclude this work.

► We develop a combinatorial technique of dealing with growth functions of three-dimensional hyperbolic polytope reflection groups. ► We study the interplay between Salem and Pisot numbers being growth rates of such groups. ► We generalise the previous result concerning hyperbolic polygon reflection groups and scope higher dimensions.