Growth rates of cocompact hyperbolic Coxeter groups and 2-Salem numbers

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 considerin…

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.

From geometry to arithmetic of compact hyperbolic Coxeter polytopes

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

Coxeter group in Hilbert geometry

A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $Γ$, find the maximal $Γ$-invariant convex, when there is a unique $Γ$-invariant convex, when the convex $Ω$ is strictly convex, when we can find a $Γ$-…

粗粒度立方体刚性

Coarse cubical rigidity

研究显示，许多右角艺术和科克塞特群的所有紧致立方体划分在粗粒度上看起来相同，诱导相同的粗粒度中位结构。该研究为非超几何群提供了此类性质的第一个例子。

CAT(0) 与立方化的 Shephard 群

CAT(0) and cubulated Shephard groups

本文将 Coxeter 群的 CAT(0) 性质推广到具有足够有限抛物子群的 Shephard 群，并证明在相应 Coxeter 群为 (FC) 型时，该群在 CAT(0) 立方复形上真且余紧作用，同时引入 A_3 单形复形为 CAT(1) 的新判据。