Compact hyperbolic tetrahedra with non-obtuse dihedral angles

Details Compact hyperbolic tetrahedra with non-obtuse dihedral angles Compact hyperbolic tetrahedra with non-obtuse dihedral angles

2006-01-07

arxiv.org/abs/math/0601148v1

Mathematics

Geometric Topology

19 pages

Scientific paper

Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles $A_C$ obtained by restricting to {\em non-obtuse} angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the space of dihedral angles of compact hyperbolic tetrahedra, after restricting to non-obtuse angles, is non-convex. Our proof provides a simple example of the ``method of continuity'', the technique used in classification theorems on polyhedra by Alexandrow \cite{ALEX}, Andreev \cite{AND}, and Rivin-Hodgson \cite{RH}.

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