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

Abstract

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.

Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

“…Based on this point of view, Kolpakov [Kol,Proposition 3 and Theorem 5] proved the following results, which generalise Floyd's work [F] from the planar to the spatial case.…”

“…If a Coxeter polyhedron P ∞ has a 4-valent ideal vertex, then Vinberg [Vi2,p. 238] indicated the following degeneration feature which was proved in detail by Kolpakov [Kol,Proposition 2]. Proposition 1.…”

Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers

“…In order to prove Theorem 3, we use the following deformation argument for Coxeter polyhedra introduced by Kolpakov in [7]. We present it in a modified form which is more suitable for further account.…”

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

“…The actual conjecture describes a detailed distribution of the poles of the associated growth series, and it implies that the growth rate is a Perron number. Several results confirming the latter fact have appeared recently in [19,20,25,30,31,32].…”

“…The original conjecture by Kellerhals and Perren has been confirmed in several cases [19,20,25,30] by applying Steinberg's formula [29] and with extensive use of hyperbolic geometry, notably Andreev's theorem [1,2]. Recently in [31,32] it was established that the growth rates of all 3-dimensional hyperbolic Coxeter groups are Perron numbers.…”