Question 3.3 — Is there a theory of random walks on Outer space similar to that of Kaimanovich and Masur for Teichm¨ uller space?

Definitions and mathematical statement

Outer space in the usual rank-$n\geq2$ model consists of marked finite core metric graphs of rank $n$, with positive edge lengths and total volume $1$, and carries the $\operatorname{Out}(F_n)$-action. A standard orbit-walk setup chooses a probability measure $\mu$ on $\operatorname{Out}(F_n)$, independent increments $\phi_i$ with that law, products $\Phi_k=\phi_1\cdots\phi_k$, and orbit points $\Phi_kx_0$ in Outer space.

The question asks for a random-walk theory analogous to Kaimanovich--Masur's Teichmüller-space theory. The excerpt does not specify the probability law, moment/entropy or non-elementarity assumptions, metric, compactification or which analogous conclusions are sought. Boundary convergence, hitting measures or a Poisson-boundary identification would each need precise additional hypotheses; none is asserted for all measures merely from this analogy.

The following sentence suggests bounded cohomology, modeled on Bestvina--Fujiwara's mapping-class subgroup work [6], as an approach to "Question 17." That numbered question is absent and is not identified with the present one, so the remark is retained without inventing a bounded-cohomology theorem or a dependency. The supplied summary reports developed random-outer-automorphism theory and mentions Rivin and others, but supplies no exact comparison theorem or complete reference.

Short English statement

Develop an Outer-space analogue of the intended Kaimanovich--Masur random-walk theory, with probability, boundary and conclusion hypotheses specified; the separate Question 17 reference remains unresolved.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 3.3, supplied PDF page 333. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Kaimanovich and Masur, named Teichmüller-space analogy; Bestvina and Fujiwara [6], bounded-cohomology/subgroup template. Full bibliography and Question 17 are absent. The summary additionally names Rivin and others without complete citations.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000283$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports substantial related progress.