Orders of elements in finite quotients of Kleinian groups

A positive integer $m$ will be called a {\it finitistic order} for an element $\gamma$ of a group $\Gamma$ if there exist a finite group $G$ and a homomorphism $h:\Gamma\to G$ such that $h(\gamma)$ has order $m$ in $G$. 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 $m>2$ 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

Real-analytic, volume-preserving actions of lattices on 4-manifolds

We prove that any real-analytic, volume-preserving action of a lattice $\Gamma$ in a simple Lie group with \Qrank(\Gamma)\geq 7 on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.Comment: 5 page

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

Dehn surgery, homology and hyperbolic volume

If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.Comment: This is the version published by Algebraic & Geometric Topology on 8 December 200

The diameter of the set of boundary slopes of a knot

Let K be a tame knot with irreducible exterior M(K) in a closed, connected, orientable 3--manifold Sigma such that pi_1(Sigma) is cyclic. If infinity is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted d_K, is a numerical invariant of K. We show that either (i) d_K >= 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [Comment. Math. Helv. 74 (1999) 530-547] with a result about the effect of cabling on boundary slopes.Comment: This is the version published by Algebraic & Geometric Topology on 29 August 200

Euler characteristics, lengths of loops in hyperbolic 3-manifolds, and Wilson's Freiheitssatz

Let $p$ be a point of an orientable hyperbolic $3$-manifold $M$, and let $m\ge1$ and $k\ge2$ be integers. Suppose that $\alpha_1,\ldots,\alpha_m$ are loops based at $p$ having length less than $\log(2k-1)$. We show that if $G$ denotes the subgroup of $\pi_1(M,p)$ generated by $[\alpha_1],\ldots,[\alpha_m]$, then $\overline{\chi}(G)\doteq-\chi(G)\le k-2$; here $\chi(G)$ denotes the Euler characteristic of the group $G$, which is always defined in this situation. This result is deduced from a result about an arbitrary finitely generated subgroup $G$ of the fundamental group of an orientable hyperbolic $3$-manifold. If $\Delta$ is a finite generating set for $G$, we define the $index\ of\ freedom$${\rm iof}(\Delta)$to be the largest integer$k$such that$\Delta$contains$k$elements that freely generate a rank-$k$free subgroup of$G$. We define the$minimum\ index\ of\ freedom$${\rm miof}(G)$to be$\min_{\Delta }{\rm iof}(\Delta )$, where$\Delta $ranges over all finite generating sets for$G$. The result is that$\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