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Hyperbolic volume, homology rank, and the Four Color Theorem

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Peter Shalen (UIC)

Abstract:
It is a consequence of the Margulis lemma that the rank of the fundamental group of a closed hyperbolic 3-manifold is bounded linearly in terms of the volume of the manifold. In particular, for any prime p there is a constant lambda such that for any closed hyperbolic 3-manifold M we have dim H_1(M;\ZZ_p\ZZ) \le lambda vol M. Agol, Leininger and Margalit showed that for any p one can take lambda=334.08.
In this talk I will discuss some improvements on this bound that are developed in a series of joint papers by Rosemary Guzman and me. For an arbitrary p we show that one can take lambda=168.602, and for p=2 one can take lambda=157.831. The proofs of these results draw on a variety of techniques from several branches of mathematics. In order to illustrate this I will focus on a step in the argument for p=2 that involves an application of the Four Color Theorem.