Collaring Theorems and the Volumes of Symmetric Hyperbolic $n$-Manifolds

F.W. Gehring, T.H. Marshall and G.J. Martin


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.


Math Review Classification
Primary 30F40, 20H10, 57N10

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26 pages

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