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Free Kleinian groups and volumes of hyperbolic 3-manifolds. (English) Zbl 0860.57011

This paper continues a line of investigation begun by Culler and Shalen on the explicit relationship between the topology, in particular the first Betti number, and the volume of a closed, orientable hyperbolic 3-manifold \(N\). The main result of the paper is a qualitative analog of the Margulis lemma for free, topologically tame, \(k\)-generator subgroups of \(\text{Isom}^+({\mathbf H}^3)\) without parabolics. Specifically, let \(\Phi\) be such a group freely generated by \(\xi_1,\dots, \xi_k\); then, for any \(z\in{\mathbf H}^3\), \[ \sum^k_{i=1} {1\over {1+\exp(\text{dist} (z,\xi_i\cdot z))}}\leq {1\over2}. \] In particular, \(\text{dist} (z,\xi\cdot z)\geq \log(2k-1)\) for some \(i\). The primary application of the main result is to the study of volume estimates arising from topological hypotheses on a closed orientable hyperbolic 3-manifold \(N\). For example, if \(\beta_1(N)\) is at least 4 and \(\pi_1(N)\) does not contain a subgroup isomorphic to the fundamental group of a closed surface of genus 2, then \(N\) contains an embedded hyperbolic ball of radius \({1\over2} \log(5)\). By work of Meyerhoff and Böröczky on volume estimates from density estimates for hyperbolic sphere packings, this implies that \(N\) has volume greater than 3.08. This has qualitative implications for the determination of the topology of the least volume closed hyperbolic 3-manifold, conjectured to be the Weeks manifold.
Other results include a lower bound on the volume of \(N\) in terms of the length of short geodesics. In particular, if the first Betti number of \(N\) is at least 3, there is a bound on the volume of \(N\) which goes to \(\pi\) as the length of the shortest geodesic goes to zero. Combining this result with Dehn surgery techniques, it is shown that a non-compact hyperbolic 3-manifold with first Betti number at least 4 has volume at least \(\pi\).

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)

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