6.1 (I. Kapovitch) — Random walks and generic pseudo-Anosov singularities

Definitions and mathematical statement

Let $S$ be a surface in the intended mapping-class-group setting, and let $\operatorname{Mod}(S)$ denote isotopy classes of its self-homeomorphisms, with orientation and marking conventions to be specified. Given a probability measure $\nu$ on this group, a random walk started at the identity is \[ W_n=X_1\cdots X_n,\qquad X_1,X_2,\ldots\ \text{independent with law }\nu. \] A pseudo-Anosov mapping class has transverse invariant stable and unstable measured foliations, scaled by reciprocal factors $\lambda$ and $\lambda^{-1}$ with $\lambda>1$. An interior foliation singularity is trivalent when it has exactly three prongs. For a closed orientable surface, the generic principal-stratum interpretation asks that all singularities of both invariant foliations have three prongs. Punctures and boundary components require separate conventions.

Problem (I. Kapovitch). Show that the random walk gives a pseudo-Anosov mapping class with generic trivalent invariant-foliation singularities with probability tending to one as its length tends to infinity. With the closed-surface interpretation just described, the requested limit is \[ \lim_{n\to\infty}\Pr_\nu\left(W_n\text{ is pseudo-Anosov and every singularity of its stable and unstable foliations is three-pronged}\right)=1. \] The extracted statement does not specify $S$, the law $\nu$, its support, moment or generation hypotheses, or whether the assertion is for a particular simple random walk or for a class of walks. Thus the display is a precise target once those data are supplied, not a claim for every probability law on every mapping class group. In particular, the needed nondegeneracy assumptions cannot be omitted. If punctures are allowed, the source must say whether "trivalent" concerns only interior singularities and what prongs are permitted at punctures.

The pseudo-Anosov conclusion alone is weaker than the asserted singularity-pattern conclusion. The registry summary attributes probability tending to one for the former to random-walk results and leaves the latter unverified.

Short English statement

Prove that a sufficiently long mapping-class-group random walk produces a pseudo-Anosov whose invariant foliations have only the intended generic three-pronged singularities, with probability tending to one. The surface and random-walk hypotheses are missing from the excerpt.

Sources

• Local UnsolvedMath registry, permanent ID 10900045, source code catalogue entry; attributed exactly as supplied to I. Kapovitch, no proposal year supplied.
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 6.1 (I. Kapovitch), PDF page 5; https://arxiv.org/pdf/1512.04620#page=5 and https://arxiv.org/abs/1512.04620. The provenance records source-TeX extraction checked against rendered PDF text and access on 2026-07-29.
• The research summary refers to Maher, "Exponential decay in the mapping class group," and related observations of Rivin on random Heegaard/gluing maps. It supplies no precise bibliographic identifiers or theorem hypotheses for these references.