Coxeter群的一个有效的Compactness定理
An Effective Compactness Theorem for Coxeter Groups
摘要：
通过高度非建设性的方法，Bestelf、Culler、Feighn、Morgan、Paulin、Rips、Shalen和Thurston的作品表明，如果一个群不分裂为一个几乎可解的子群，那么它在双曲n-空间上的离散和忠实作用的空间，模共轭，在所有维度上都是紧的。虽然这意味着这样的群的双曲结构的空间具有有限直径，但已知的方法没有给出明确的界限。我们为Coxeter群建立了这样一个界。我们发现，要么组分裂的一个虚拟的可解子群或有一个常数C和一个点在双曲n-空间移动不超过C的任何发电机。常数C只取决于群的生成元的个数，而与关系子无关。
英文摘要：
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.
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