Problem 1.1 — Investigate the dependence of Expρ,⟨f,M ⟩ on the marked Riemann surface ⟨f,M⟩.

Definitions and mathematical statement

In the usual Teichmüller convention, a marked Riemann surface $\langle f,M\rangle$ consists of a Riemann surface $M$ and a marking $f:S\to M$ from a fixed oriented reference surface. Markings are identified by a biholomorphism compatible with them up to the prescribed homotopy. The source does not state the reference surface type or its boundary/puncture conventions.

Problem 1.1 asks to investigate how the source's object \[ \operatorname{Exp}_{\rho,\langle f,M\rangle} \] depends on the marked Riemann surface $\langle f,M\rangle$, with the other parameter $\rho$ held fixed. No definition of $\operatorname{Exp}$ or $\rho$, nor the object's domain, codomain, equivalence relation, topology or normalization, is supplied. Thus the request concerns a source-defined family over marked Riemann surfaces, but no claim of independence, continuity, differentiability, holomorphicity or another specified dependence property can be inferred.

In particular, the notation must not be silently interpreted as an ordinary matrix exponential or as a chosen geometric exponential map. The registry's character-variety/Goldman context does not determine the missing construction.

Short English statement

Study the variation of the source's $\operatorname{Exp}_{\rho,\langle f,M\rangle}$ as the marked Riemann surface varies. The operator and representation parameter are undefined in the excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000185$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 1.1, PDF page 211: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• The registry mentions character-variety/Goldman theory but supplies no precise construction, author attribution for this question, or additional authoritative citation.