Die Dehnsche Zerlegungsinvariante für hyperbolische Polyederbausteine. (The Dehn partition invariant for hyperbolic orthoschemes.) (Q919280)

The author gives an elementary computation for the Dehn invariant \(\psi (L(\alpha))=\log | 2 \sin \alpha | \otimes \alpha\) of the doubly asymptotic orthoscheme L(\(\alpha\)) \((0<\alpha <\pi /2)\) in the hyperbolic space \(H^ 3\). This orthoscheme L(\(\alpha\)) is represented in the Poincaré half space model by the vertices \(A_ 0(\infty)\), \(A_ 1(0,0,1)\), \(A_ 2(\cos^ 2\alpha,\sin \alpha \cos \alpha,\sin \alpha)\), \(A_ 3(1,0,0)\); the faces are perpendicular to the plane \(z=0\), respectively \(A_ 1A_ 2A_ 3\) lies on the unit hemisphere of centre (0,0,0). The Dehn invariant of a convex polyhedron B is \(\psi (B)=\sum_{K}\ell_ K\otimes \alpha_ K\) where the sum runs over the edges K of B, \(\ell_ K\) denotes the length of K, \(\alpha_ K\) is the dihedral angle reduced modulo \(\pi\) /2 at the edge K. This definition can naturally be extended for asymptotic polyhedra in \(H^ 3\) by taking horospheres about the infinite vertices. The author nicely motivates his topic and the relations with volume and scissors congruence problems in \(H^ 3\). The basic reference is \textit{J. L. Dupont} and \textit{C.-H. Sah}, J. Pure Appl. Algebra 25, 159-195 (1982; Zbl 0496.52004) and ibid. 30, 217 (1983).