Kirby Problem 2.5

Definitions and mathematical statement

Let $S_g$ be a closed connected oriented surface of genus $g\geq3$. Its mapping class group acts on $H_1(S_g;\mathbb Z)\cong\mathbb Z^{2g}$, preserving the algebraic intersection pairing. The Torelli group is the kernel \[ \mathcal I_g=\ker\left(\operatorname{Mod}(S_g) \longrightarrow\operatorname{Sp}_{2g}(\mathbb Z)\right), \] so its elements act trivially on first integral homology. A finite presentation of a group is a description $\langle x_1,\ldots,x_m\mid r_1,\ldots,r_k\rangle$ with finitely many generators and relator words, presenting that group exactly.

For each $g\geq3$, find a finite presentation of $\mathcal I_g$, or prove that no finite presentation exists. A positive answer must prove that the stated relators generate all relations among the chosen generators. Providing a finite generating set alone does not answer this question. A negative answer must exclude every possible finite presentation, rather than just show that a particular natural presentation has infinitely many relators. The genus range is part of the question; the low-genus Torelli groups are not substitutes for the groups specified here.

Short English statement

Is the Torelli group finitely presented in each genus at least three, and can such presentations be made explicit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2753 (KP-2.5), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.5. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.