Free subgroups of surface mapping class groups

Anderson, James A.; Aramayona, Javier; Shackleton, Kenneth J.

doi:10.1090/s1088-4173-07-00156-7

Cited by 5 publications

(6 citation statements)

References 24 publications

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“…We thank Dick Canary for pointing out the error in [1]. We also thank Dick Canary and Juan Souto for conversations.…”

Section: Acknowledgementsmentioning

confidence: 90%

“…However, up to conjugation, there are only finitely many pairs of independent pseudo-Anosov mapping classes of translation distance at most L and whose axes are at distance at most D, by a result of Ivanov (stated as Theorem 2.1 in [1]) and the discreteness of MCG(Σ). Therefore, we may choose a R 0 = R 0 (L, D, Σ) that works for any two independent pseudo-Anosov mapping classes of translation distance at most L, whose axes are at distance at most D, as desired.…”

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confidence: 99%

“…One then obtains correct versions of Corollaries 4.4, 4.5, 5.5 and 5.6, and Theorem 5.7 of [1] by replacing R with R 0 and considering, instead of -precompact geodesics, axes of two pseudo-Anosov mapping classes whose translation distances are bounded above by a fixed constant and whose axes are within a fixed bounded distance of one another. We remark that the corrected version of Theorem 5.7 follows from Corollary 4.7 and the fact that there are only finitely many pairs of independent pseudo-Anosov mapping classes of translation distance at most L and whose axes are at distance at most D.…”

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confidence: 99%

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Corrigendum to “Free subgroups of surface mapping class groups”

Anderson

Aramayona

Shackleton

2009

Conform. Geom. Dyn.

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Abstract. We provide a corrigendum to the results of Conform. Geom. Dyn. 11 (2007), 44-55, pointing out an error in the proofs of Propositions 4.3 and 5.4 and providing corrected statements.The proofs of Propositions 4.3 and 5.4 in [1] are incorrect: controlling the length of the path q does not imply that q is a quasi-geodesic, and thus part (3) of Theorem 4.2 cannot be applied here. Consequently the estimate for the constant R given in the statements of Proposition 5.4, Corollary 5.5, Corollary 5.6, and Theorem 5.7 does not apply.In Proposition A below, we offer a corrected version of Proposition 5.4, for a constant R 0 that still depends only on the translation distances of the pseudoAnosovs and the distance between their axes. However, we add that the method of proof does not allow us to obtain an explicit estimate for R 0 , unlike that quoted in Proposition 5.4.Proposition A also recovers Proposition 4.3 sufficiently to deduce the generation of free subgroups, treating the pertinent case of geodesic axes for pseudo-Anosov mapping classes. This said, the result on generation of free subgroups we are able to obtain is now a somewhat straightforward consequence of the work of Ivanov. It seems to be unknown whether Proposition A holds for all precompact Teichmüller geodesics.Subsequent to the publication of [1] more general results regarding the generation of free subgroups of the mapping class groups were obtained by Fujiwara Proof. Let λ ∈ Fix (φ) and µ ∈ Fix (ψ). Since φ and ψ are independent, λ and µ are distinct and together fill Σ. There exists a unique biinfinite Teichmüller geodesic from λ to µ, which we denote by [λ, µ]. Let M = M (φ, ψ) > 0 be such that

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“…We thank Dick Canary for pointing out the error in [1]. We also thank Dick Canary and Juan Souto for conversations.…”

Section: Acknowledgementsmentioning

confidence: 90%

mentioning

confidence: 99%

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confidence: 99%

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Corrigendum to “Free subgroups of surface mapping class groups”

Anderson

Aramayona

Shackleton

2009

Conform. Geom. Dyn.

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“…D. Long shows in [41][Lemma 2.6] that a normal subgroup N of the mapping class group of a closed surface of genus at least three (the genus hypothesis comes in to show that the mapping class group in question has trivial center) has at least two non-commuting pseudo-anosov mapping classes. Once we know that, a standard ping-pong argument (see, e.g., [3]) shows that N contains a free group on two generators, and so is not solvable. Theorem 4.14.…”

Section: Lemma 43 There Is No Pair Of Extensionsmentioning

confidence: 99%

“…These results can be viewed as analogous (in the complex/Kähler category) to the classical Stallings fibration Theorem, which states that: Theorem 1.3 (Stallings Fibration Theorem, [50]). A compact irreducible 3-manifold M 3 fibers over S 1 if and only if π 1 (M 3 ) admits a surjection onto Z with finitely generated kernel.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of Fibering

Rivin¹

2011

Preprint

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Given a manifold M, it is natural to ask in how many ways it fibers (we mean fibering in a general way, where the base might be an orbifold -this could be described as Seifert fibering)There are group-theoretic obstructions to the existence of even one fibering, and in some cases (such as Kähler manifolds or three-dimensional manifolds) the question reduces to a group-theoretic question. In this note we summarize the author's state of knowledge of the subject.I would like to thank Ilya Kapovich for much enlightenment on geometric group theory, D. Arapura for supplying the proof of Theorem 1.2, Igor Belegradek and Alain Valette on putting me abreast of the recent developments on groups with finite outer automorphism group, Fred Cohen for interesting discussions on different sorts of monodromy, N. Katz and P. Sarnak, who piqued the author's interest in monodromy, and the Institute for Advanced Study for its hospitality, which made this work possible. I would like to thank Igor Belegradek, Tom Church, and David Futer on comments on a previous version of this preprint -in particular, Belegradek had pointed out the very interesting preprint [22]. I would also like to thank

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