An Effective Compactness Theorem for Coxeter Groups

TL;DR

This paper establishes an explicit bound on the displacement of generators in hyperbolic space for Coxeter groups, linking group splitting properties to geometric bounds.

Contribution

It provides a new, explicit compactness bound for Coxeter groups in hyperbolic space, improving upon previous non-constructive results.

Findings

Either the Coxeter group splits over a virtually solvable subgroup or a universal displacement bound exists.

The displacement bound depends only on the number of generators, not on relators.

The result offers a concrete criterion for geometric actions of Coxeter groups.

Abstract

Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Paulin, 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 hyperbolic n-space, 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 hyperbolic n-space 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.