283 research outputs found
On the growth of cocompact hyperbolic Coxeter groups
For an arbitrary cocompact hyperbolic Coxeter group G with finite generator
set S and complete growth function P(x)/Q(x), we provide a recursion formula
for the coefficients of the denominator polynomial Q(x) which allows to
determine recursively the Taylor coefficients and the pole behavior of the
growth function of G in terms of its Coxeter subgroup structure. We illustrate
this in the easy case of compact right-angled hyperbolic n-polytopes. Finally,
we provide detailed insight into the case of Coxeter groups with at most 6
generators, acting cocompactly on hyperbolic 4-space, by considering the three
combinatorially different families discovered and classified by Lanner,
Kaplinskaya and Esselmann, respectively.Comment: 24 page
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 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.Comment: 26 pages, 16 figures, 4 data tables; minor corrections; European
Journal of Combinatorics, 201
The growth function of Coxeter garlands in
The growth function W(t) of a Coxeter group W relative to a Coxeter generating set is always a rational function. We prove by an explicit construction that there are infinitely many cocompact Coxeter groups W in hyperbolic 4-space with the following property. All the roots of the denominator of W(t) are on the unit circle except exactly two pairs of real root
- …