Boundaries of Groups and Kleinian Groups — Problem 72

Definitions and mathematical statement

Let $S$ be a finite-type surface in the intended mapping-class setting, let $\mathcal T(S)$ be its Teichmüller space, and let $\operatorname{Mod}(S)$ act by changing markings. Write $\operatorname{PMF}(S)$ for measured foliations modulo multiplication of transverse measure by a positive scalar. The surface type is not specified in the excerpt.

Given a probability law $\mu$ on $\operatorname{Mod}(S)$, take independent increments $g_i$ with law $\mu$, put $w_n=g_1\cdots g_n$, and fix $x_0\in\mathcal T(S)$. Under hypotheses ensuring almost-sure boundary convergence $w_nx_0\to\xi\in\operatorname{PMF}(S)$, the hitting measure is \[ \nu(B)=\mathbb P(\xi\in B) \quad\text{for Borel }B\subseteq\operatorname{PMF}(S). \] The problem asks to characterize this measure and in particular whether it is absolutely continuous relative to visual measure. If $\sigma_{x_0}$ denotes the intended Lebesgue measure class on the sphere of directions at $x_0$, transported to $\operatorname{PMF}(S)$, the precise absolute-continuity question is \[ \sigma_{x_0}(B)=0\quad\Longrightarrow\quad\nu(B)=0 \qquad\text{for every Borel }B. \] This asks $\nu\ll\sigma_{x_0}$, not mutual absolute continuity.

In a standard Teichmüller formulation the direction sphere is the unit sphere of holomorphic quadratic differentials at $x_0$, with the identification through their measured foliations. The excerpt gives no basepoint, normalization, or exact visual-measure construction. It also gives no step law, support/non-elementarity assumptions, moment hypotheses, or surface restrictions guaranteeing the stated limit. Whether the question is for a specific walk, all walks in a class, or a criterion on $\mu$ remains unspecified, so no unrestricted universal statement is inserted.

The summary attributes harmonic/hitting-measure results to Kaimanovich--Masur (1996) and reports only partial progress on the comparison with visual measure, mentioning Masur's measure. It provides no exact theorem or identification of that measure with the specified visual class. These are source attributions, not independently verified current status.

Short English statement

Describe the boundary hitting distribution of a random walk by mapping classes, and decide whether it is absolutely continuous with respect to the intended visual Lebesgue class. The surface, walk hypotheses, basepoint, and visual-measure convention are not specified in the excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6200072$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Moon Duchin (2005), Problem 72 in Kapovich, Problems on Boundaries of Groups and Kleinian Groups, PDF page 19: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; current status marked as needing review.
• Kaimanovich--Masur (1996) and Masur's measure are mentioned in the summary without full citations or exact hypotheses.