Generating the genus \(g+1\) Goeritz group of a genus \(g\) handlebody (Q2867801)

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 \textit{E. Akbas} [Pac. J. Math. 236, No. 2, 201--222 (2008; Zbl 1157.57002)]. For higher genus cases, \textit{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 \textit{M. Scharlemann} [Bol. Soc. Mat. Mex., III. Ser. 10, 503--514 (2004; Zbl 1095.57017)].NEWLINENEWLINEThe 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.NEWLINENEWLINEFor the entire collection see [Zbl 1272.57002].