Finite-Sided Weil–Petersson Dirichlet Domains

Definitions and mathematical statement

Let $S$ denote the surface intended in the source, let $\mathcal T(S)$ be its Teichmüller space equipped with the Weil--Petersson metric, and let $(X,d_{\mathrm{WP}})$ be its metric completion. Write $\Gamma=\operatorname{Mod}(S)$ for the intended mapping class group, acting on $X$ by the extensions of its Weil--Petersson isometries. The source does not specify the topological type of $S$, conventions concerning punctures or boundary components, or the precise mapping class group.

For a center $p\in X$, define the closed Dirichlet region \[ D_p=\bigcap_{\gamma\in\Gamma}\left\{x\in X: d_{\mathrm{WP}}(x,p)\leq d_{\mathrm{WP}}(x,\gamma p)\right\}. \] Problem. Does there exist a center $p$ for which $D_p$ is a finite-sided Dirichlet fundamental domain for the action of $\Gamma$ on $X$? The original source wording asks to show that some such domain exists.

One precise interpretation of the finite-sided requirement is that there exist a center $p$ and a finite subset $F\subseteq\Gamma$ such that $D_p$ is a fundamental domain and \[ D_p=\bigcap_{\gamma\in F}\left\{x\in X: d_{\mathrm{WP}}(x,p)\leq d_{\mathrm{WP}}(x,\gamma p)\right\}. \] This is a conditional interpretation, not a definition confirmed by the supplied source: the record does not define a side or establish that counting geometric sides is equivalent to finite determination by these inequalities. It also does not specify whether the center must lie in $\mathcal T(S)$, rather than in its completion, or how the fundamental-domain convention treats stabilizers and shared boundary points. The surface scope and these conventions require clarification before the question has a unique fully quantified formulation.

Short English statement

The question asks whether the mapping class group has a Dirichlet fundamental domain with finitely many sides in the Weil--Petersson completion of Teichmüller space. The supplied record leaves the surface type, permitted centers, and precise meaning of finite-sidedness unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003947$, title: Finite-Sided Weil--Petersson Dirichlet Domains; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2018$. Its proposer field is empty; a supplied reference label identifies this as the Wolpert problem.
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, pp. 2475--2534, DOI 10.4171/owr/2018/40. https://doi.org/10.4171/owr/2018/40. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Oberwolfach Report 40, Wolpert problem." https://doi.org/10.4171/owr/2018/40.
• Supplied literature reference: "Wolpert, WP completion geometry." https://arxiv.org/abs/math/0502528.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the finite-sided Dirichlet-domain question remains unresolved, and that later work on Weil--Petersson completion and CAT(0) geometry does not establish the required finite-sided fundamental domain. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.