Question — Which properties of the braid groups can be extended to the mapping class groups?

Definitions and mathematical statement

Research question. Which properties of braid groups extend to mapping class groups? Here $B_n$ denotes the braid group on $n$ strands, and the mapping class group depends on the chosen surface and on orientation, boundary, and marked-point conventions.

The source poses a comparative program, not one universally quantified theorem about every formal property. A precise instance must identify a braid-group property, the relevant strand range, the target surface class, and whether an analogue, a literal transfer, or a specified strengthening is sought. The excerpt gives no chosen properties, transfer map, target genus range, or criterion for completion. Thus no fixed list of positive or negative examples, exhaustive classification, or decision algorithm is invented.

The supplied research summary mentions residual finiteness and linearity and asserts that many properties fail to extend, including a claim of mapping-class-group nonlinearity. It provides no hypotheses or supporting citations for those assertions. They are retained as unverified editorial suggestions, not mathematical facts or answers to the source question.

Short English statement

Investigate which specified braid-group properties have analogues for specified mapping class groups. The source gives a broad research program; the summary's unsupported examples do not supply a verified classification.

Sources

• Local UnsolvedMath registry, permanent ID 11000329, source code catalogue entry, unnumbered Question, PDF page 386. 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 Problem, Conjecture, and Question headings across chapters, with repeated labels disambiguated by page.
• No separate authoritative reference or precise property comparison is supplied. The research summary mentions residual finiteness and linearity/nonlinearity without theorem hypotheses or citations; those claims require review.
• Supplied status open and OPEN-TRIAGE. The summary calls this a broad program with no single answer. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.