On extremal properties of hyperbolic Coxeter polytopes and their reflection groups
type: Dataset
description: This thesis concerns hyperbolic Coxeter polytopes, their reflection groups and associated combinatorial and geometric invariants. Given a Coxeter group $G$ realisable as a discrete subgroup of $\mathrm{Isom}\,\mathbb{H}^n$, there is a fundamental domain $\mathscr{P} \subset \mathbb{H}^n$ naturally associated to it. The domain $\mathscr{P}$ is a Coxeter polytope. Vice versa, given a Coxeter polytope $\mathscr{P}$, the set of reflections in its facets generates a Coxeter group acting on $\mathbb{H}^n$. The reflections give a natural set $S$ of generators for the group $G$. Then we can express the growth series $f_{(G,S)}(t)$ of the group $G$ with respect to the generating set $S$. By a result of R.~Steinberg, the corresponding growth series is the power series of a rational function. The growth rate $\tau$ of $G$ is the reciprocal to the radius of convergence of such a series. The growth rate is an algebraic integer and, by a result of J.~Milnor, $\tau > 1$. By a result of W.~Parry, if $G$ acts on $\mathbb{H}^n$, $n=2,3$, cocompactly, then its growth rate is a Salem number. By a result of W.~Floyd, there is a geometric correspondence between the growth rates of cocompact and finite co-volume Coxeter groups acting on $\mathbb{H}^2$. This correspondence gives a geometric picture for the convergence of Salem numbers to Pisot numbers. There, Pisot numbers correspond to the growth rates of finite-volume polygons with ideal vertices. We reveal an analogous phenomenon in dimension $3$ by considering degenerations of compact Coxeter polytopes to finite-volume Coxeter polytopes with four-valent ideal vertices. In dimension $n\geq 4$, the growth rate of a Coxeter group $G$ acting cocompactly on $\mathbb{H}^n$ is known to be neither a Salem, nor a Pisot number. A particularly interesting class of Coxeter groups are right-angled Coxeter groups. In the case of a right-angled Coxeter group acting on $\mathbb{H}^n$, its fundamental domain $\mathscr{P} \subset \mathbb{H}^n$ is a righ…
issued: 2026-06-01T12:09:44.031676
language: en
modified: 2026-06-01T12:09:44.031683
contact point: Organization
  fn: CCSD
keyword: coxeter-group
keyword: groupe-de-coxeter
keyword: growth-rate
keyword: growth-series
keyword: hyperbolic-polytope
keyword: infoeu-reposemanticsdoctoralthesis
keyword: mathmath-comathematics-mathcombinatorics-mathco
keyword: mathmath-grmathematics-mathgroup-theory-mathgr
keyword: mathmath-mgmathematics-mathmetric-geometry-mathmg
keyword: mathmath-ntmathematics-mathnumber-theory-mathnt
keyword: nombre-de-pisot
keyword: nombre-de-salem
keyword: pisot-number
keyword: polyedre-hyperbolique
keyword: salem-number
keyword: serie-daccroissement
keyword: taux-daccroissement
keyword: theses
landing page: https://theses.hal.science/tel-00762315

On extremal properties of hyperbolic Coxeter polytopes and their reflection groups
type: Distribution
format: HTML
issued: 2026-06-01T12:09:44.205054
modified: 2026-06-01T12:09:43.969957
access URL: https://theses.hal.science/tel-00762315

test_moissonnage_selune
type: Agent