65 research outputs found
Orders of elements in finite quotients of Kleinian groups
A positive integer will be called a {\it finitistic order} for an element
of a group if there exist a finite group and a
homomorphism such that has order in . It is
shown that up to conjugacy, all but finitely many elements of a given finitely
generated, torsion-free Kleinian group admit a given integer as a
finitistic order.Comment: 21 pp. I have largely rewritten Section 2 in order to correct the
statement of Proposition 2.7. The original statement was not logically clear,
and was not well adapted to an application in the more recent paper [22
Singular surfaces, mod 2 homology, and hyperbolic volume, II
If M is a closed simple 3-manifold whose fundamental group contains a genus-g
surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least
max(3g-1,6), we show that M contains a closed, incompressible surface of genus
at most g. This improves the main topological result of part I, in which the
the same conclusion was obtained under the stronger hypothesis that the
dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M
is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then
H_1(M;Z_2) has dimension at most 5.Comment: 23 pages. This version incorporates suggestions from the referee and
adds a new section giving examples showing that the main theorem is almost
sharp for genus 2. The examples have mod 2 homology of rank 4 and their
fundamental groups contain genus 2 surface groups, but they have no closed
incompressible surface
Margulis numbers for Haken manifolds
For every closed hyperbolic Haken 3-manifold and, more generally, for any
hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken
manifold, the number 0.286 is a Margulis number. If M has non-zero first Betti
number, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis
number for M.Comment: 25 pages. Some statements were clarified some typos were corrected
and some of the propositions were generalize
Euler characteristics, lengths of loops in hyperbolic 3-manifolds, and Wilson's Freiheitssatz
Let be a point of an orientable hyperbolic -manifold , and let
and be integers. Suppose that are
loops based at having length less than . We show that if
denotes the subgroup of generated by
, then ;
here denotes the Euler characteristic of the group , which is
always defined in this situation.
This result is deduced from a result about an arbitrary finitely generated
subgroup of the fundamental group of an orientable hyperbolic -manifold.
If is a finite generating set for , we define the $index\ of\
freedom{\rm iof}(\Delta)k\DeltakkGminimum\ index\ of\ freedom{\rm miof}(G)\min_{\Delta
}{\rm iof}(\Delta )\Delta G\overline{\chi}(G)<{\rm iof}(G)$. The author has
recently learned that this is equivalent to a special case of a theorem about
arbitrary finitely presented groups due to J. S. Wilson.Comment: In this version, which is 14 pages long, I have added a preface
explaining that Theorem B is a special case of a theorem due to J. S. Wilson.
I have modified the title, abstract and bibliography accordingl
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