Kirby Problem 2.1

Definitions and mathematical statement

Let $S$ be a connected orientable surface of finite type and genus $g\geq3$, with finitely many punctures and boundary components allowed. Its mapping class group $\operatorname{Mod}(S)$ consists of isotopy classes of orientation-preserving homeomorphisms, fixing boundary points and preserving the puncture set; isotopies obey the same conditions. A subgroup $G$ has finite index if there are finitely many cosets of $G$ in $\operatorname{Mod}(S)$. Its abelianization is \[ G^{\mathrm{ab}}=G/[G,G], \] where $[G,G]$ is generated by commutators $xyx^{-1}y^{-1}$.

The Ivanov conjecture asks whether $G^{\mathrm{ab}}$ is finite for every such surface and every finite-index subgroup $G$. These groups are finitely generated, so the question is equivalently whether $H_1(G;\mathbb Q)=0$, or whether no such $G$ admits a nonzero homomorphism to $\mathbb Z$. The finite abelian quotient may depend on the subgroup; there is no proposed common upper bound for its order. Fixing punctures individually instead changes to a finite-index subgroup and gives the same universal finite-index formulation.

Short English statement

Must every finite-index subgroup of a mapping class group of genus at least three have finite abelianization?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2749 (KP-2.1), 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 2.1. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.