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Generating the genus $$g+1$$ Goeritz group of a genus $$g$$ handlebody. (English) Zbl 1288.57014
Hodgson, Craig D. (ed.) et al., Geometry and topology down under. A conference in honour of Hyam Rubinstein, Melbourne, Australia, July 11–22, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8480-5/pbk). Contemporary Mathematics 597, 347-369 (2013).
For the $$3$$-sphere, the genus $$g$$ Goeritz group consists of the isotopy classes of orientation-preserving homeomorphisms of the $$3$$-sphere that leave the genus $$g$$ Heegaard splitting invariant. When $$g=2$$, its finite presentation is known by E. Akbas [Pac. J. Math. 236, No. 2, 201–222 (2008; Zbl 1157.57002)]. For higher genus cases, J. Powell [Trans. Am. Math. Soc. 257, 193–216 (1980; Zbl 0445.57008)] gave a set of generators, but his proof contained a gap, pointed out by M. Scharlemann [Bol. Soc. Mat. Mex., III. Ser. 10, 503–514 (2004; Zbl 1095.57017)].
The purpose of the paper under review is to give a finite set of generators of the genus $$g+1$$ Goeritz group $$G(H,\Sigma)$$ of a genus $$g\;(\geq 1)$$ handlebody $$H$$. Here, $$\Sigma$$ is a genus $$g+1$$ Heegaard surface of $$H$$. Indeed, a concrete set of $$4g+1$$ generators is described. If $$g\geq 2$$, then $$G(H,\Sigma)$$ is shown to be isomorphic to the fundamental group of the space $$\mathrm{Unk}(I,H)$$ of unknotted arcs in $$H$$. When $$g=1$$, there is a surjection from $$\mathrm{Unk}(I,H)$$ to $$G(H,\Sigma)$$. The author exhibits two proofs, using classical techniques and thin position.
For the entire collection see [Zbl 1272.57002].

##### MSC:
 57M40 Characterizations of the Euclidean $$3$$-space and the $$3$$-sphere (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
##### Keywords:
Goeritz group; handlebody; thin position
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