Collineation groups of translation planes of small dimension.
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...