Kirby Problem 2.2

Definitions and mathematical statement

Let $S$ be a connected oriented surface of finite type, with punctures allowed, and write $\operatorname{Mod}(S)$ for its orientation-preserving mapping class group: homeomorphisms modulo isotopy. In this formulation there are no boundary circles fixed pointwise; equivalently, boundary components are treated as punctures. Put $\pi=\pi_1(S)$. A subgroup $K\leq\pi$ is characteristic if every automorphism of $\pi$ preserves $K$. For a finite-index characteristic $K$, the action of mapping classes on $\pi$ gives a homomorphism \[ \rho_K:\operatorname{Mod}(S)\longrightarrow\operatorname{Out}(\pi/K), \qquad \Gamma(K)=\ker\rho_K. \] Here $\operatorname{Out}(Q)$ is the automorphism group of $Q$ modulo inner automorphisms; using it removes the choice of basepoint. Call a subgroup congruence if it contains some $\Gamma(K)$.

The congruence subgroup problem asks whether every finite-index subgroup $H\leq\operatorname{Mod}(S)$ is congruence. Thus the requested assertion is that for every such $H$ there exists a finite-index characteristic subgroup $K$ of $\pi$ with $\Gamma(K)\subseteq H$. The finite quotients $\pi/K$ may be arbitrary; restricting to homology modulo an integer would be a different, stronger requirement.

Short English statement

Does every finite-index subgroup of a surface mapping class group contain the kernel of its action on a suitable finite quotient of the surface group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2750 (KP-2.2), 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 2.2. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.