On commensurable hyperbolic Coxeter groups
type: Item
instance of: scholarly article
title: On commensurable hyperbolic Coxeter groups
zbMATH Open document ID: 1378.51011
author: Rafael Guglielmetti
author: Matthieu Jacquemet
author: Ruth Kellerhals
published in: Geometriae Dedicata
publication date: 2016-08-23T00:00:00Z
full work available at URL: kel chc.pdf
review text: Let \(n\in {\mathbb N}\) and \(f : {\mathbb R}^{n+1}\times {\mathbb R}^{n+1}\rightarrow {\mathbb R}\), \(f(x,y):=x_1y_1+\dots+x_ny_n - x_{n+1} y_{n+1}\). Then, \(({\mathbb R}^{n+1}, f)\) is the (real) Minkowski space and \({\mathbb H}^n :=\{ x \in {\mathbb R}^{n+1} ~:~f(x,x)=-1\) and \(x_{n+1} > 0\}\) together with the metric given by \(f\) is (a model of) the \(n\)-dimensional hyperbolic space. Each anisotropic vector \(e\in {\mathbb R}^{n+1}\) (i.e. \(f(e,e)\not= 0\)) supplies the reflection \(\sigma _e\) at the hyperplane \(H=e^\perp\), i.e. the orthogonal transformation \(\in\mathrm O({\mathbb R}^{n+1},f)\) that maps \(e\) to \(-e\) and \(v\) to \(v\) for each vector \(v\in e^{\perp}\). The group \(\mathrm{Isom}({\mathbb H}^n )\) consists of the orthogonal mappings \(\alpha \in\mathrm O({\mathbb R}^{n+1},f)\) such that \(\alpha ({\mathbb H}^n )={\mathbb H}^n \). A subgroup \(\Gamma\) of \(\mathrm{Isom}({\mathbb H}^n )\) is called a (geometric) hyperbolic Coxeter group if \(\Gamma\) is a discrete group generated by finitely many reflections at hyperplanes \(H_i\). As in the Euclidean case, a hyperbolic Coxeter group provides a fundamental domain \(P\) which is a polyhedron \(P=\bigcap _{i=1}^N {H_i}^-\) (with \(H_i=e_i^\perp\) and \(H_i^- := \{ v\in {\mathbb H}^n, ~f(x,e_i)\leq 0 \}\)) where the angles at the intersections of pairs of faces are integer submultiples of \(\pi\) (or zero). Hyperbolic Coxeter groups \(\Gamma\), \(\Gamma'\) are called commensurable if \(g^{-1}\Gamma g \cap \Gamma '\) has finite index in both \(\Gamma\) and \(\Gamma '\) for some \(g\in\mathrm{Isom}({\mathbb H}^n )\). Commensurability is an equivalence relation preserving properties such as cocompactness, cofiniteness and arithmeticity. The article studies Coxeter groups generated by \(N = n+2\) reflections at hyperplanes bounding a pyramid \(P\subseteq {\mathbb H}^n\) of finite volume such that the neighborhood of the apex is a product of two simplices of positive dimensions. These gro…
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51B20
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 6618889
zbMATH Keywords: commensurability
zbMATH Keywords: hyperbolic Coxeter group
zbMATH Keywords: Coxeter pyramid
zbMATH Keywords: amalgamated free product
zbMATH Keywords: translational lattice
zbMATH Keywords: arithmetic group
describes a project that uses: CoxIter
MaRDI profile type: Publication
OpenAlex ID: W2322144095
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DOI: 10.1007/S10711-016-0151-7
description: scientific article; zbMATH DE number 6618889