Question — Do there exist any other examples of non commuting homeomorphisms g andh which are not both Dehn twists and satisfy a…

Definitions and mathematical statement

For elements $g,h$ of a group, write $W_m(g,h)$ for the alternating word with exactly $m$ letters beginning with $g$; for example, $W_3(g,h)=ghg$ and $W_4(g,h)=ghgh$. A braid relation of length $m$ is $W_m(g,h)=W_m(h,g)$.

Question. Are there other examples of noncommuting surface homeomorphisms $g,h$, not both Dehn twists, satisfying a braid relation of some integer length $m=m_{i,j}>2$? In the intended ambient group this asks for \[ gh\neq hg,\qquad W_m(g,h)=W_m(h,g),\qquad m\geq3, \] with $g,h$ not simultaneously single Dehn twists. The condition permits one of them to be a Dehn twist; it is not strengthened to excluding twists individually. The surface, boundary/marking conventions, previously known examples referred to by "other," and whether equality is in the homeomorphism group or only in the mapping class group $M(S)$ are unspecified. Those two equality notions are different.

The source suggests using positive examples to seek an embedding associated to $E_6$, an embedding of a group with a triangular Dynkin diagram, or an interesting homomorphism of a non-small Artin group into $M(S)$ with nonabelian image. The exact meaning of $E_6$, the triangle labels, "non-small," and equivalence of constructions are not given; no Artin-group embedding follows merely from one two-generator braid relation. For reducible homeomorphisms it speculates that twists along boundary curves would again have to satisfy braid relations, returning to a geometric or almost-geometric embedding. This is stated as a proposed heuristic, not a proved reduction or a restriction excluding reducible examples.

Short English statement

Find noncommuting surface maps, not both single Dehn twists, satisfying an alternating braid relation of length at least three. The source suggests larger Artin-group constructions as a goal, but leaves the surface, equality convention, and previously known examples unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 11000146, source code catalogue entry, unnumbered "Question," PDF page 131. 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 labels disambiguated by page.
• The supplied summary suggests connections to "Lefevre / Meyer" conjectures on abstract braid-relation generators, but provides no precise conjecture, reliable identification, or full citation. No such identification is adopted.
• Supplied status open and OPEN-TRIAGE. The summary says not settled without a theorem or status audit. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.