An Effective Compactness Theorem for Coxeter Groups

Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a virtually solvable subgroup, then the space of its discrete and faithful actions on Hn, modulo conjugation, is compact for all dimensions. Although this implies that the space of hyperbolic structures of such groups has finite diameter, the known methods do not give an explicit bound. We establish such a bound for Coxeter groups. We find that either the group splits over a virtually solvable subgroup or there is a constant C and a point in Hn that is moved no more than C by any generator. The constant C depends only on the number of generators of the group, and is independent of the relators. INTRODUCTION The space of discrete and faithful actions of a given group G on Hn, up to conjugation, is a deformation space of the group. It is denoted D(G, n). In the 1980’s, Thurston proved that when a group G is the fundamental group of an orientable, compact, irreducible, acylindrical 3-manifold with boundary, the deformation space D(G, 3) is compact [Thu86]. To prove this result, Thurston analysed sequences of ideal triangulations. Inspired by Thurston’s work and Culler-Shalen’s work on varieties of threemanifold groups [CS83], Morgan-Shalen reproved Thurston’s compactness theorem using methods from algebraic geometry and geometric topology [MS84] [MS88a] [MS88b]. Morgan then showed that when G is the fundamental group for a compact, orientable, and irreducible 3-manifold, the space D(G, n) is compact if and only if the group G does not admit a virtually abelian splitting [Mor86]. This result was pushed to include all finitely-presented groups using the Rips Machine by Bestvina-Feighn, who state the following Compactness Theorem as a consequence of the main result of [BF95] concerning actions of trees: Compactness Theorem for Finitely Presented Groups (Thurston, Morgan-Shalen, Morgan, Rips, Bestvina-Feighn). If G be a finitely-presented group that is not virtually abelian and does not split over a virtually solvable subgroup, then D(G, n) is compact. If a finitely-presented group does not split over a virtually solvable subgroup, then the Compactness Theorem implies that there is a point in Hn that is not moved too far by any generator, for any action by the group. However, the methods in [BF95] and [Mor86] do not give an explicit bound. The technical adjective ineffective describes such non-constructive results. In contrast, if a proof is constructive or yields explicit quantities, then it is termed effective. The main result of 2000 Mathematics Subject Classification. 20F65 (Primary) 57M99 (Secondary). This material is based upon work supported by the National Science Foundation Grants DMS0135345, DMS-05-54349, DMS-04-05180, and DMS-0602191. 1 ar X iv :0 90 2. 27 18 v1 [ m at h. G T ] 1 6 Fe b 20 09