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A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. (English) Zbl 1157.57002

Let \({\mathcal H}_2\) be the group of isotopy classes of orientation-preserving homeomorphisms of \(S^3\) that preserve the isotopy class of an unknotted handlebody of genus \(2\). By a theorem of Goeritz (1933) it is known that \({\mathcal H}_2\) is finitely-generated. That fact was subsequently proved again by M. Scharlemann [Topology Appl. 156, No. 2, 165–185 (2004; Zbl 1095.57017)], whose approach involved constructing a connected simplicial \(2\)-complex \(\Gamma\) on which \({\mathcal H}_2\) acts, and which deformation retracts onto a graph \(\tilde{\Gamma}\). In this paper, the author uses Bass-Serre theory to show that the graph \(\tilde{\Gamma}\) is a tree, in which shortest paths can be calculated algorithmically. As a consequence, the following finite presentation for \({\mathcal H}_2\) is obtained:
\({\mathcal H}_2 \cong \langle\, A,B,C,D \;| \;A^2 = C^2 = D^3 = (AC)^2 = [A,B] = [A,D] = 1, \, [B,C] = A, \, [D,C] = D \, \rangle\). Accordingly, \({\mathcal H}_2\) is a free product of \(\langle B,A,C \rangle \cong (\mathbb{Z} \oplus \mathbb{Z}_2) \rtimes \mathbb{Z}_2\) and
\(\langle D,C,A \rangle \cong (\mathbb{Z}_3 \rtimes \mathbb{Z}_2) \oplus \mathbb{Z}_2\) with a Klein 4-subgroup \(\langle A,C \rangle \cong\mathbb{Z}_2 \oplus \mathbb{Z}_2\) amalgamated.

MSC:

57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 1095.57017
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