NDLI: Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

Content Provider	Springer Nature Link
Author	Sakuma, Makoto Ohshika, Ken’ichi
Copyright Year	2015
Abstract	Let $$M=H_1\cup _S H_2$$ be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup $${\text {MCG}}^0(H_j)$$ of the mapping class group of $$H_j$$ consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of $$H_j$$ homotopic to the identity, and let $$G_j$$ be the subgroup of the automorphism group of the curve complex $$\mathcal {CC}(S)$$ obtained as the image of $${\text {MCG}}^0(H_j)$$ . Then the group $$G=\langle G_1, G_2\rangle $$ generated by $$G_1$$ and $$G_2$$ acts on $$\mathcal {CC}(S)$$ with each orbit being contained in a homotopy class in M. In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R-bounded combinatorics and has high Hempel distance.
Ending Page	134
Page Count	18
Starting Page	117
File Format	PDF
ISSN	00465755
e-ISSN	15729168
Journal	Geometriae Dedicata
Issue Number	1
Volume Number	180
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-06-18
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Topological methods in group theory Geometric structures on low-dimensional manifolds Kleinian groups Fundamental groups and their automorphisms Heegaard splitting Projective lamination space Mapping class group
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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