Problem 12 — Calculate various group-theoretic boundaries of mapping class and Torelli groups, for instance, Tits boundaries.

Definitions and mathematical statement

Problem 12. Calculate various group-theoretic boundaries of mapping class groups and Torelli groups, including, for example, Tits boundaries. The excerpt does not specify a genus range, a surface convention, a boundary construction, or a metric space on which either group acts.

In the conventional CAT(0) setting, for a chosen geodesic CAT(0) space $X$, the visual boundary consists of geodesic rays modulo asymptotic equivalence, and the Tits boundary equips those endpoints with the path metric associated to the angular metric. Thus writing $\partial_T X$ requires the space and its metric. A proposed group-boundary calculation must identify the model $X$, the group action, the relevant boundary equivalence, and whether the requested result concerns topology, Tits metric, or equivariant structure. This explanation does not assume that a mapping class or Torelli group admits a specified proper cocompact CAT(0) action, or that a Tits boundary is canonically determined by its abstract group.

The request includes both mapping class and Torelli groups and allows boundary theories besides the example of Tits boundaries. It is not replaced by the Gromov boundary of a curve complex or another chosen auxiliary space, which would require its own comparison to the intended group-theoretic boundary. No single quantified classification theorem can be recovered until the boundary models and equivalences are fixed.

Short English statement

Compute boundary invariants for mapping class and Torelli groups, with Tits boundaries offered as an example. The source omits the metric models and boundary conventions, so a canonical boundary or a specific CAT(0) action cannot be presumed.

Sources

• Local UnsolvedMath registry, permanent ID 11000142, source code catalogue entry, Problem 12, PDF page 121. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The edited volume contains 329 explicitly typeset headings across chapters, with repeated numbering disambiguated by page.
• The supplied summary mentions studied mapping-class boundaries and a name rendered "Klarrich," without a precise result or full reference. No bibliographic correction or identification is invented.
• Supplied status open and OPEN-TRIAGE. The summary calls full Tits-boundary programs open, without specifying a model or theorem. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.