Problem 3.5 — Is every finite rank subgroup of Whittlesey’s group a Schottky subgroup of MCG (S)?

Definitions and mathematical statement

Let $W\leq\operatorname{MCG}(S)$ be the particular Whittlesey group intended by the source. The question is whether every finite-rank subgroup $H\leq W$ is Schottky in $\operatorname{MCG}(S)$.

In the usual mapping-class convention, a Schottky subgroup is a finitely generated free convex cocompact subgroup: its orbit in Teichmüller space is quasiconvex in the intended metric. If $W$ is being used as a free group, finite rank means finitely many free generators. The construction of $W$, surface $S$, exact convex-cocompactness definition and treatment of rank-zero/rank-one groups are not supplied; they must be fixed to interpret the universal quantifier. No passage to powers of generators or to a smaller subgroup is allowed as a substitute for $H$ itself being Schottky, and no uniform quasiconvexity constant over all $H$ is requested.

The continuation gives a consequence of its Theorem 2.1: if $H$ has a finite-index Schottky subgroup, then the associated extension $\Gamma_H$ has a finite-index word-hyperbolic subgroup, hence is itself word hyperbolic. The usual extension has the form \[ 1\longrightarrow\pi_1(S)\longrightarrow\Gamma_H\longrightarrow H\longrightarrow1. \] Its precise construction and Theorem 2.1 hypotheses, especially the surface convention, are missing. This is reported background about virtually Schottky groups, not the stronger answer that all finite-rank subgroups of $W$ are Schottky.

Short English statement

Is every finite-rank subgroup of the specified Whittlesey group already a free convex cocompact mapping-class subgroup, with the associated virtual-Schottky extension consequence kept separate?

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 3.5 and continuation, supplied PDF page 272. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Whittlesey's group and the source Theorem 2.1 are named without their definitions, full statements or separate bibliography entries in this record.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000233$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports unverified status.