Mapping Class Groups of Heegaard Splittings

Johnson, Jesse; Rubinstein, Hyam

doi:10.1142/s0218216513500181

Cited by 16 publications

(13 citation statements)

References 39 publications

(60 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, by work of Namazi [Nam07] and Johnson [Joh10], it turned out that the Goeritz group is always a finite group if the distance of the Heegaard splitting is at least 4. This is completely different from the case for the splittings of distance at most 1, where the Goeritz group is always an infinite group (see Johnson-Rubinstein [JR13] and Namazi [Nam07]). The notion of distance can naturally be defined for bridge decompositions as well, see for example Bachman-Schleimer [BS05].…”

Section: Introductionmentioning

confidence: 79%

Goeritz groups of bridge decompositions

Hirose

Iguchi

Kin

et al. 2020

Preprint

View full text Add to dashboard Cite

For a bridge decomposition of a link in the 3-sphere, we define the Goeritz group to be the group of isotopy classes of orientationpreserving homeomorphisms of the 3-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. This gives an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the 3-sphere and the real projective space.

show abstract

Section: Introductionmentioning

confidence: 79%

Goeritz groups of bridge decompositions

Hirose

Iguchi

Kin

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The mapping class groups of Heegaard splittings have been studied extensively and can be quite complicated. For example, when g > k, the group M od(# k S 1 × S 2 , Σ g ) will always have pseudo-Anosov elements [6].…”

Section: Distances Of Trisectionsmentioning

confidence: 99%

Comparing 4–manifolds in the pants complex via trisections

Islambouli

2018

Algebr. Geom. Topol.

View full text Add to dashboard Cite

Given two smooth, oriented, closed 4-manifolds M 1 and M 2 , we construct two invariants, D P (M 1 , M 2 ) and D(M 1 , M 2 ), coming from distances in the pants complex and the dual curve complex respectively. To do this, we adapt work of Johnson on Heegaard splittings of 3-manifolds to the trisections of 4-manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of "nearby" manifolds. This naturally leads to various graphs of 4-manifolds coming from unbalanced trisections, and we briefly explore their properties. arXiv:1707.07108v1 [math.GT]

show abstract

“…It is easy to see that Remark 3.1. There are surface bundles of arbitrarily high genus which have genus two Heegaard splittings [8]. If M(F, ϕ) contains a strongly irreducible Heegaard surface H, then d(ϕ) ≤ −χ(H) [5].…”

Section: Preliminariesmentioning

confidence: 99%