Problem 7 — (Algorithm for Veech groups).

Definitions and mathematical statement

In the customary finite-area setting, a translation surface is a surface with flat coordinate changes $z\mapsto z+c$ away from finitely many singularities, equivalently a suitable pair $(X,\omega)$ with an abelian differential. An orientation-preserving affine self-map has a constant linear derivative, and its derivatives form the Veech group \[ \Gamma(X,\omega)=\{D a:a\in\operatorname{Aff}^+(X,\omega)\}\le\mathrm{SL}_2({\mathbb{R}}). \] For a quadratic differential, the flat changes can be $z\mapsto\pm z+c$; the derivative is then naturally defined modulo sign, giving the projective Veech group in $\mathrm{PSL}_2({\mathbb{R}})$. The area, singularity/pole, and sign conventions need to follow the intended surface class.

Problem 7 (algorithm for Veech groups). Is there an algorithm that determines the full Veech group of a general translation surface or quadratic differential? A precise algorithmic formulation requires an effective encoding of the input, such as polygonal gluing data with specified effectively represented periods, and a stated kind of exact group description as output. These are not supplied in the excerpt. Arbitrary real coordinates cannot be treated as finite exact input without a computation convention. Likewise, determining a group could mean a membership decision procedure, a generator description, or a fundamental-domain construction; these are not identical tasks. No general finite-generation hypothesis or finite presentation is imposed by the source.

The source highlights square-tiled surfaces as an interesting class of Veech surfaces: they are obtained by gluing finitely many equal-sized squares. This is a named special case, not a restriction of the general algorithm request. The research summary reports algorithms in some cases but supplies no precise references or scope for them.

Short English statement

Find an exact algorithm for the Veech group of an arbitrary translation surface or quadratic differential, beyond special cases such as square-tiled surfaces. Specify the input encoding and what description of the possibly infinite group must be returned.

Sources

• Local UnsolvedMath registry, permanent ID 11000214, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 7, PDF page 259; 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.
• Source labels [Zo3] and [HuLe1] for square-tiled surfaces; full bibliography entries are not reproduced. The research summary supplies no individual citations for later algorithms.