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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Ken’ichi Ohshika

  2. Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

    Makoto Sakuma

Authors
  1. Ken’ichi Ohshika
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  2. Makoto Sakuma
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Correspondence to Ken’ichi Ohshika.

Additional information

The first author was supported by JSPS Grants-in-Aid 22654008. The second author was supported by JSPS Grants-in-Aid 21654011.

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Ohshika, K., Sakuma, M. Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions. Geom Dedicata 180, 117–134 (2016). https://doi.org/10.1007/s10711-015-0094-4

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