On Closed Leaves of Foliations, Multisections and Stable Commutator Lengths

Bowden, Jonathan

doi:10.1142/s1793525311000696

Cited by 6 publications

(6 citation statements)

References 16 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While our results recover some known computations of second bounded cohomology (see e.g. [30,38,6,26]), the previous proofs of these facts rely on a careful analysis of second cohomology, and in doing so they often appeal to deep theorems which are specific to their setting. The advantage of our approach is that it bypasses the cohomological techniques and only focuses on the bounded part.…”

Section: Introductionsupporting

confidence: 71%

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Fournier-Facio¹,

Lodha²

2021

Preprint

View full text Add to dashboard Cite

We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of 1-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group G 0 constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to the so-called homological von Neumann-Day Problem, as discussed by Calegari. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Next, we provide the first examples of finitely generated left orderable groups that are not locally indicable, and yet have vanishing second bounded cohomology. This proves that a homological analogue of the Witte-Morris Theorem does not hold. Finally, we combine some of the aforementioned results to provide the first examples of manifolds whose simplicial volumes are algebraic and irrational. This is further evidence towards a conjecture of Heuer and Löh.

show abstract

Section: Introductionsupporting

confidence: 71%

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Fournier-Facio¹,

Lodha²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…While circulating an earlier version of this article, we found out that such an adjunction bound in the case of surface bundles of fiber and base genera g, h ≥ 2 was independently obtained by Bowden [3], where the author studies multisections of surface bundles.…”

Section: Let [S]mentioning

confidence: 73%

Sections of surface bundles and Lefschetz fibrations

Baykur

Korkmaz

Monden

2013

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h − 2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g ≥ 2.

show abstract

“…A sharp upper bound on the self-intersection number of a section of a surface bundle over a surface was originally obtained by Morita in [14], and was later obtained using gauge theory in [2] and [4], respectively. This bound constitutes one side of the inequality leading to the calculation of the precise commutator length of the boundary parallel Dehn twist in Map(Σ 1 g ) [2].…”

Section: Upper Bounds On the Self-intersection Numbers Of Sectionsmentioning

confidence: 99%

“…In [2] it was moreover shown that a similar upper bound holds for Lefschetz fibrations if we remove the absolute value; a proof of which, for sections that miss the critical locus, can be obtained using the above approach and following the framework of [16] for example.) The proof in [2] as well as the one in [4] uses gauge theory, appealing to Seiberg-Witten invariants on minimal symplectic 4manifolds and the adjunction inequality for Seiberg-Witten basic classes. As seen from the above discussion however, the use of gauge theory is rather superfluous.…”

Section: Upper Bounds On the Self-intersection Numbers Of Sectionsmentioning

confidence: 99%

Flat bundles and commutator lengths

Baykur¹

2012

Preprint

View full text Add to dashboard Cite

The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary Σ of genus⌋ in the mapping class group Map(Σ). The alternative proof we provide goes through push maps and Morita's use of Milnor-Wood inequalities, in particular it does not appeal to gauge theory. In turn, we produce infinite families of pairwise non-homotopic 4-manifolds admitting genus g surface bundles over genus h surfaces with distinguished sections which are flat but admit no flat connections for which the sections are flat, for every fixed pairs of integers g, h ≥ 2. The latter result generalizes a theorem of Bestvina, Church, and Souto, and allows us to obtain a simple proof of Morita's non-lifting theorem (for an infinite family of non-conjugate subgroups) in the case of marked surfaces.

show abstract

On Closed Leaves of Foliations, Multisections and Stable Commutator Lengths

Cited by 6 publications

References 16 publications

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Sections of surface bundles and Lefschetz fibrations

Flat bundles and commutator lengths

Contact Info

Product

Resources

About