Problem 2.3 — (Volume spectrum).

Definitions and mathematical statement

Fix the genus $g$ and the preceding source's representation space $X_g(\Gamma)$ and volume maps $\operatorname{Vol}_k$. For each integer $1\leq k\leq3g-3$, determine \[ \mathcal V_{g,k}(\Gamma) =\{\operatorname{Vol}_k(\rho):\rho\in X_g(\Gamma)\}\subset{\mathbb{R}}, \qquad \bigcup_{\Gamma\ \mathrm{finitely\ presented}} \mathcal V_{g,k}(\Gamma). \] The first request fixes $\Gamma$; the second varies over all finitely presented groups. Also determine the images when one restricts to representations having holomorphic representatives or representatives totally geodesic with respect to a fixed metric, and how those geometric conditions constrain possible volumes.

The excerpt omits the definitions of $X_g(\Gamma)$, $\operatorname{Vol}_k$, the geometric representatives and their domains. Consequently no integral, normalization or homology pairing is substituted for the named volume map. The source notes Mirzakhani's recursive Weil--Petersson volume formulas for moduli of surfaces with nonempty totally geodesic boundary; these are contextual moduli-space volumes, not the requested representation-volume image.

A following discussion proposes further computations from linear representations. In the usual conjugacy-class interpretation of $X_g(\Gamma)$, for $\rho:\Gamma\to\operatorname{Mod}_g$, $\psi:\operatorname{Mod}_g\to\operatorname{GL}_m({\mathbb{C}})$, a conjugation-invariant function $c$ and a fixed $\gamma\in\Gamma$, composition gives the invariant $ [\rho]\mapsto c(\psi(\rho(\gamma)))$. The source also proposes subgroup commensurability invariants obtained from such representations, cites computations in [Su2], and calls for more. It reports that no faithful linear representation was known for $g\geq3$; this is retained as a dated source assertion, not a current literature conclusion. The construction of commensurability invariants is not supplied.

Short English statement

Determine the images of each representation-volume map, their union over finitely presented groups, and the restrictions imposed by holomorphic or totally geodesic representatives; the volume definitions are missing from the extract.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 2.3 ("Volume spectrum"), supplied PDF page 23, with the following linear-representation discussion. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. The supplied running heading names "Some problems on mapping class groups and moduli space.' '
• The source cites Mirzakhani [Mir] for boundary Weil--Petersson recursion and [Su2] for linear-representation computations; complete bibliography entries are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000010$, exact source code catalogue entry; supplied OPEN-TRIAGE summary distinguishes the representation-volume question from Mirzakhani's recursion and reports no resolution.