Problem 2.7 — Study the handlebody subgroup of Mg.

Definitions and mathematical statement

Let $H_g$ be a genus-$g$ handlebody with a fixed boundary identification $\partial H_g\cong S_g$. In the usual oriented convention its handlebody subgroup is \[ \mathcal H_g=\{[f]\in\operatorname{Mod}(S_g): f\text{ extends to a homeomorphism of }H_g\}. \] The symbol $M_g$ in the source denotes the ambient mapping class group here, not moduli space. The precise orientation, marking, and allowed genus conventions are not given.

Problem 2.7. Study $\mathcal H_g\leq M_g$. In particular, find a simplified presentation revealing new structural information, and/or new information about its coset representatives in $M_g$. A presentation should identify generators and defining relations for the intended subgroup, but the source gives no existing presentation to simplify, no criterion for simplicity, and no requirement of a particular finite size. For the coset question, the intended left or right cosets and equivalence of representatives must be specified; no canonical transversal or algorithmic normal form is asserted to exist or silently demanded. The wording "and/or" permits either direction and is retained. This is a research program rather than a single conjectural identity or a prescribed full classification.

Short English statement

Investigate the subgroup of surface mapping classes that extend over a handlebody, especially through a simpler structural presentation or new information about coset representatives. The source leaves presentation and coset conventions open.

Sources

• Local UnsolvedMath registry, permanent ID 11000164, source code catalogue entry, Problem 2.7, PDF page 157. No individual proposer is supplied; the page distinguishes repeated labels in the volume.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across contributed chapters.
• The supplied summary mentions handlebody-subgroup work, Tamagawa, and unspecified other authors, but provides no theorem, presentation, or full bibliographic entry.
• Supplied status open and OPEN-TRIAGE. The summary describes known structure and an ongoing area rather than a precise completed result. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.