Problem 5.4 — Suppose that G <MCG (S) is a finite co-area Veech subgroup.

Definitions and mathematical statement

Let $G<\operatorname{MCG}(S)$ be a finite-coarea Veech subgroup. In the customary interpretation, $G$ is induced by affine symmetries of a translation/half-translation surface, preserving a Teichmüller disk, with derivative Fuchsian group having finite hyperbolic covolume. Any derivative kernel or projectivization convention must agree with the source.

Problem 5.4. Describe the group denoted $\Gamma_C$ in the source for such a $G$. In particular, does $\Gamma_C$ contain the associated group $\Gamma_G$ as a subgroup of finite index? \[ \Gamma_G\le\Gamma_C,\qquad [\Gamma_C:\Gamma_G]<\infty. \] This is the stated index direction: the larger group is $\Gamma_C$. The excerpt does not define $C$, $\Gamma_C$, $\Gamma_G$, or the intended inclusion, so the display is a precise rendering of its target conditional on those missing constructions, not an independent definition of them.

The surrounding chapter concerns surface-group extensions. In the standard closed-surface formulation, for $\Pi=\pi_1(S)$ and a mapping-class subgroup $G\le\operatorname{Out}(\Pi)$, its pullback in $\operatorname{Aut}(\Pi)$ gives \[ 1\longrightarrow\Pi\longrightarrow\Gamma_G\longrightarrow G\longrightarrow1. \] The adjacent local item 11000240 relates $\Gamma_C$ to fiber-respecting quasi-isometries of $\Gamma_G$, meaning quasi-isometries that coarsely preserve the family of cosets of the surface-group kernel. This identifies the intended coarse-geometric context, but does not supply the missing definition of $C$ or establish which precise identification applies in the present Veech case. It would be unjustified to choose an arbitrary subgroup called $C$, replace it by a Fuchsian commensurator, or infer the requested finite-index statement just from finite coarea.

Short English statement

For a finite-coarea Veech subgroup, determine the source's associated coarse-geometric group $\Gamma_C$, especially whether the surface-group extension $\Gamma_G$ has finite index in it. The construction of $C$ is missing from the excerpt.

Sources

• Local UnsolvedMath registry, permanent ID 11000239, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, chapter context "Problems in the geometry of surface group extensions," Problem 5.4, PDF page 275; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• Adjacent local ID 11000240 supplies the fiber-respecting quasi-isometry context and source label [Mos03a], but does not reproduce the definition of $C$. Full bibliography and theorem statements are absent from the present record.