| Let $Gamma$ be a discrete subgroup of orientation preserving isometries of hyperbolic $n$-space, and let ${cal F}$ be a collection of finite subgroups of $Gamma$, for which any two distinct fixed point sets of groups in ${cal F}$ are disjoint. Under some minor additional hypotheses we prove that, for some $r>0$, the tabular neighbourhoods of radius $r$ about these sets, called collars, are also pairwise disjoint. As an application we show that if $F$ is a finite subgroup of $Gamm$ which has a precisely invariant fixed lpoint set of dimension $d$ and such that the orbit of any point not in the fixed point set contains at least $p$ points, then the radius $r$ is bounded below by a value depending only on $n, d$ and $p$, and which goes to infinity as $p$ does. We use these results to obtain lower bounds for volumes of hyperbolic $n$-manifolds whose symmetry groups have nonempty fixed point sets. |
Keywords
Math Review Classification
Primary 30F40, 20H10, 57N10
Last Updated
24/3/97
Length
26 pages
Availability
This article is available in: