| Through highly non-constructive methods, works by Bestvina, Culler, Morgan, Paulin, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a small subgroup, then the space of its discrete and faithful actions on hyperbolic n-space, modulo conjugation, is compact for all n. We make this result effective for Coxeter groups. By fixing a Coxeter presentation and associating a finite simplicial tree to a given action, we find that either the group splits over a small subgroup or there is a constant C and a point in hyperbolic n-space that is moved no more than C by any generator. The constant C depends only on n and the number of generators in the presentation.;We begin by recalling Coxeter groups and special graphs of groups decompositions. Following this, we discuss geometry used to prove the main result: we review some hyperbolic geometry, define augmented Gromov approximating trees, show that they are quasi-isometric to quasi-convex hulls in hyperbolic n-space, and construct a "shadow" of such a tree in hyperbolic n-space. We then move from a geometric setting to "labelling systems", a combinatorial framework which we use to build special graph of groups decompositions. Finally, we use the above to estimate C . |
