Collineation Groups Whose Order is Prime to the Characteristic.
We consider face-to-face partitions of bounded polytopes into convex polytopes in for arbitrary and examine their colourability. In particular, we prove that the chromatic number of any simplicial partition does not exceed . Partitions of polyhedra in into pentahedra and hexahedra are - and -colourable, respectively. We show that the above numbers are attainable, i.e., in general, they cannot be reduced.
Let be a building of arbitrary type. A compactification of the set of spherical residues of is introduced. We prove that it coincides with the horofunction compactification of endowed with a natural combinatorial distance which we call the root-distance. Points of admit amenable stabilisers in and conversely, any amenable subgroup virtually fixes a point in . In addition, it is shown that, provided is transitive enough, this compactification also coincides with the group-theoretic...
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 RE. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles AC obtained by restricting to 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...