[Paper Review] The genus two Goeritz group of $S^2 imes S^1$

Sangbum Cho, Yuya Koda|arXiv (Cornell University)|Mar 28, 2013

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes a finite presentation for the genus-2 Goeritz group of the 3-manifold $\mathbb{S}^2 \times \mathbb{S}^1$, proving it is finitely presented by constructing a tree on which the group acts with a single edge as the quotient. The explicit presentation is $\langle \epsilon \rangle \oplus \langle \alpha \mid \alpha^2 = 1 \rangle \oplus \langle \beta, \gamma, \sigma \mid \gamma^2 = \sigma^2 = (\gamma\beta\sigma)^2 = 1 \rangle$, derived via group-theoretic amalgamations of stabilizer subgroups of disk pairs in a genus-2 Heegaard splitting.

ABSTRACT

The genus-g Goeritz group is the group of isotopy classes of orientation-preserving homeomorphisms of a closed orientable 3-manifold that preserve a given genus-g Heegaard splitting of the manifold. In this work, we show that the genus-2 Goeritz group of $S^2 imes S^1$ is finitely presented, and give its explicit presentation.

Motivation & Objective

• To determine the algebraic structure of the genus-2 Goeritz group of $\mathbb{S}^2 \times \mathbb{S}^1$, a 3-manifold with a unique genus-2 Heegaard splitting up to isotopy.
• To prove that this Goeritz group is finitely presented, addressing an open problem in 3-manifold topology.
• To construct an explicit finite presentation of the group using geometric and group-theoretic techniques.

Proposed method

• The authors construct a tree on which the genus-2 Goeritz group acts with quotient a single edge, enabling the use of Bass-Serre theory for group presentations.
• They analyze stabilizer subgroups of various disk pairs in the Heegaard surface, using the action on the complex of primitive disks.
• The group is decomposed as an amalgamated free product of stabilizer subgroups of pairs of primitive disks and their common dual disks.
• Generators are identified geometrically: $\epsilon$ as a Dehn twist about $\partial E_0$, $\alpha$ as a half-twist, and $\beta, \gamma, \sigma$ as involutions or elements of order 3.
• Relations are derived from the topology of the Heegaard splitting and the intersection patterns of essential disks in the handlebodies.
• The final presentation is obtained by combining presentations of stabilizer subgroups and using the relation $\tau = \gamma\beta$ to close the amalgamation.

Experimental results

Research questions

• RQ1 Is the genus-2 Goeritz group of $\mathbb{S}^2 \times \mathbb{S}^1$ finitely presented?
• RQ2 What is the explicit finite presentation of this group in terms of geometric generators?
• RQ3 How do stabilizer subgroups of disk pairs contribute to the global group structure via amalgamated products?
• RQ4 Can the group action on a disk complex be used to derive a finite presentation?
• RQ5 What role do common dual disks and symmetric disk pairs play in the group decomposition?

Key findings

• The genus-2 Goeritz group of $\mathbb{S}^2 \times \mathbb{S}^1$ is finitely presented, resolving a previously open problem.
• The group has the explicit presentation $\langle \epsilon \rangle \oplus \langle \alpha \mid \alpha^2 = 1 \rangle \oplus \langle \beta, \gamma, \sigma \mid \gamma^2 = \sigma^2 = (\gamma\beta\sigma)^2 = 1 \rangle$, with generators realized as isotopy classes of orientation-preserving homeomorphisms.
• The stabilizer subgroup $\mathcal{G}_{\{E,E'\}}$ is presented as $\langle \epsilon \rangle \oplus \langle \beta, \beta' \mid (\beta\beta')^2 = 1, \beta\beta' = \beta'\beta \rangle$, reflecting a symmetric action on a primitive pair.
• The stabilizer $\mathcal{G}_{\{D\cup E\}}$ has presentation $\langle \epsilon \rangle \oplus \langle \alpha \mid \alpha^2 = 1 \rangle \oplus \langle \sigma, \tau \mid \sigma^2 = 1, (\tau\sigma)^2 = 1 \rangle$, arising from an amalgamated product over a common subgroup.
• The stabilizer $\mathcal{G}_{\{D,E\}}$ is presented as $\langle \epsilon \rangle \oplus \langle \tau \rangle \oplus \langle \alpha \mid \alpha^2 = 1 \rangle$, with $\tau$ acting as a swap of symmetric disks.
• The full group presentation is derived by combining these stabilizer presentations using the relation $\tau = \gamma\beta$, confirming the final structure.