Hyperbolic volume, mod 2 homology, and k-freeness

Guzman, Rosemary K.; Shalen, Peter B.

doi:10.48550/arxiv.2010.03676

Cited by 2 publications

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“…The surprising feature of the arguments in this paper is that they require an application of the Four Color Theorem. This further illustrates a pattern that was already seen in the papers [2], [3], [11], [12], and [15], in which a broad range of techniques and results from pure mathematics were brought to bear on the problem of bounding dim H 1 (M; F p ) in terms of a given bound on vol M, where M is a finite-volume orientable hyperbolic 3-manifold and p is a prime. These papers invoke the log(2k − 1) Theorem, which itself is proved using a Banach-Tarski-style decomposition of the Patterson-Sullivan measure for a free Kleinian group the Marden tameness conjecture, proved by Agol [1] and Calegari-Gabai [9].…”

Section: Introductionsupporting

confidence: 55%

“…At a few points in this paper, it will be convenient to use the following notation which was used in [15]. If p is a point of a non-simply connected hyperbolic 3-manifold M, we denote by s 1 (p) the length of the shortest homotopically non-trivial closed path in M based at p. Thus s 1 is a positive-valued function on M, and for any ε > 0 we have M thin (ε) = {p ∈ M : s 1 (p) < ε} and M thick (ε) = {p ∈ M : s 1 (p) ≥ ε}.…”

Section: 2mentioning

confidence: 99%

“…. , N; since the distance function on the metric space M is continuous, and since s 1 is continuous by [15,Lemma 4.5], we may suppose the P i to be chosen so that s 1 (p i ) > ε 4 for i = 1, . .…”

Section: 4mentioning

confidence: 99%

“…These papers invoke the log(2k − 1) Theorem, which itself is proved using a Banach-Tarski-style decomposition of the Patterson-Sullivan measure for a free Kleinian group the Marden tameness conjecture, proved by Agol [1] and Calegari-Gabai [9]. Other results and techniques invoked in the papers [2], [3], [11], [12], and [15] include Perelman's estimates for Hamilton's Ricci flow, via the applications in [5]; the tower construction used by Shapiro and Whitehead in their proof of Dehn's lemma ( [22]); the Borsuk nerve theorem; the topological theory of 3-manifold group actions on trees; and Fisher's inequality on block designs.…”

Section: Introductionmentioning

confidence: 99%

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The ratio of homology rank to hyperbolic volume, II

Guzman¹,

Shalen²

2022

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Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod p homology (for any prime p) of a finite-volume orientable hyperbolic 3-manifold M in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem.If M is closed, and either (a) π 1 (M ) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus 2, 3 or 4, or (b) p = 2, and M contains no (embedded, two-sided) incompressible surface of genus 2, 3 or 4, then dim H 1 (M ; F p ) < 157.763•vol(M ). If M has one or more cusps, we get a very similar bound assuming that π 1 (M ) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus g for g = 2, . . . , 8. These results should be compared with those of our previous paper "The ratio of homology rank to hyperbolic volume, I," in which we obtained a bound with a coefficient in the range of 168 instead of 158, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction.The arguments also give new linear upper bounds (with constant terms) for the rank of π 1 (M ) in terms of vol M , assuming that either π 1 (M ) is 9-free, or M is closed and π 1 (M ) is 5-free.

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Section: Introductionsupporting

confidence: 55%

Section: 2mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The ratio of homology rank to hyperbolic volume, II

Guzman¹,

Shalen²

2022

Preprint

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“…Proposition 5.3 is deduced from Proposition 5.2 by the same method as is used for the corresponding step in [3]. The constant that appears in Theorem 5.3 incorporates a small improvement that is provided by one of the results of [9].…”

Section: Introductionmentioning

confidence: 99%

The ratio of homology rank to hyperbolic volume, I

Guzman¹,

Shalen²

2021

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We show that for every finite-volume hyperbolic 3-manifold M and every prime p we have dim H 1 (M ; F p ) < 168.05 • vol M . This improves on a result proved by Agol, Leininger and Margalit giving the same inequality with the coefficient 168.05 replaced by 334.08, and on the analogous result with a coefficient of about 260 which would be obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if M is a finite-volume orientable hyperbolic 3-manifold such that π 1 (M ) is 2-semifree, then rank π 1 (M ) < 1 + λ 0 • vol M , where λ 0 is a certain constant less than 167.79.

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Hyperbolic volume, mod 2 homology, and k-freeness

Cited by 2 publications

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The ratio of homology rank to hyperbolic volume, II

The ratio of homology rank to hyperbolic volume, II

The ratio of homology rank to hyperbolic volume, I

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