On commensurable hyperbolic Coxeter groups
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 groups are called hyperbolic Coxeter pyramid groups and have been investigated by E. Vinberg, P. Tumarkin and others. The authors classify hyperbolic Coxeter pyramid groups up to commensurability. A criterion by E. Vinberg distiguishes arithmetic and non-arithmetic pyramid groups. The classification of arithmetic pyramid groups uses algebraic invariants and methods by \textit{C. Maclachlan} and \textit{A. W. Reid} [J. Lond. Math. Soc., II. Ser. 58, No. 3, (1998; Zbl 0922.57009); The arithmetic of hyperbolic 3-manifolds. New York, NY: Springer (2003; Zbl 1025.57001)]. The classification of non-arithmetic groups is mainly based on the field generated by powers of traces of Coxeter elements.
scientific article; zbMATH DE number 6618889
scientific article