arXiv subjects

Peter B. Shalen

Publications and source records attributed to Peter B. Shalen.

24 records · Page 2Linked to original sources

arXiv2009

Betti numbers and injectivity radii

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds 0.32798. For comparison, Andrew Przeworski showed, with no topological restrictions, that the maximal injectivity radius exceeds arcsinh(1/4) = 0.247..., while the authors showed that if M has first Betti number at least 3 then the maximal injectivity exceeds log(3)/2 = 0.549.... The proof combines a result due to Przeworski with techniques developed by the authors in the 1990s.

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arXiv2009

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the complement of an incompressible surface in M.

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arXiv2008

Volume and homology of one-cusped hyperbolic 3-manifolds

Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5 for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1, then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and that the compact core of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06.

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arXiv2008

Singular surfaces, mod 2 homology, and hyperbolic volume, I

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. This result should be viewed as an analogue of Dehn's Lemma for π_1-injective singular surfaces. The geometric application states that if M is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric theorem combines the topological theorem with several deep geometric results, including the Marden tameness conjecture,recently established by Agol and by Calegari-Gabai; a co-volume estimate for 3-tame, 3-free Kleinian groups due to Anderson, Canary, Culler and Shalen; and a volume estimate for hyperbolic Haken manifolds recently proved by Agol, Storm and W. Thurston.

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arXiv2006

Characteristic Subsurfaces, Character Varieties and Dehn Filling

We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.

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arXiv2002

Characteristic subsurfaces and Dehn filling

Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes αand βon the boundary of M. The method applies when βis the boundary slope of an essential surface F that is not a semi-fiber (i.e. F is not a fiber and does not split M into two twisted I-bundles), and the Dehn filling M(α) contains a suitable singular surface. One of the main results is that if F is planar and if the fundamental group of M(α) does not contain a non-abelian free subgroup then the intersection number of αand βis at most 5.

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