Question — Consider the Artin group A5 (the braid group on six strings) divided by the relation (a1a2a3a4)5 = a5a4a3a2a2 1a2a3a4a5.

Definitions and mathematical statement

Let $A_5=B_6$ be the six-strand braid group, with standard generators $a_1,\ldots,a_5$, relations $a_i a_{i+1}a_i=a_{i+1}a_i a_{i+1}$, and $a_i a_j=a_j a_i$ for $|i-j|\geq2$. Put \[ c=a_1a_2a_3a_4,\qquad h=a_5a_4a_3a_2a_1^2a_2a_3a_4a_5, \qquad Q=B_6/\langle\!\langle c^5h^{-1}\rangle\!\rangle. \] The flattened $a^2_1$ is read as $a_1^2$, consistent with the displayed ten-letter $h$ and the source's word lengths; the typography should be confirmed.

Question. If a positive word $w$ in $a_1,\ldots,a_5$ represents \[ u=\Delta_4^4=c^{10}\in Q, \] must its letter length be $40$, $30$, or $20$, and must its ordered factorization be Hurwitz equivalent to one of \[ \Delta_4^4=(a_1a_2a_3a_4)^{10},\qquad \Delta_5^2=(a_1a_2a_3a_4a_5)^6,\qquad h^2=(a_5a_4a_3a_2a_1^2a_2a_3a_4a_5)^2, \] respectively? Positive means no inverse letters, and powers contribute their expanded number of letters. In a standard group-factorization convention, an elementary Hurwitz move replaces $(x,y)$ by $(xyx^{-1},x)$, or its inverse. Such moves preserve the ordered product and number of factors, so the three candidate forms have their indicated distinct lengths. The source refers to [1] for its convention; whether global simultaneous conjugation or restrictions on intermediate factors are included must be checked rather than added.

The source reports an affirmative related result for a closed surface: Siebert and Tian [1] prove that a genus-two symplectic Lefschetz fibration with no reducible fibers and transitive monodromy is holomorphic. It describes the monodromy factorization as a product of conjugates of factors Hurwitz equivalent to $\Delta_5^2$ or $h^2$, and explicitly says $\Delta_4^4$ lacks transitive monodromy. It then discusses each factor becoming a boundary twist for a surface with boundary, suggesting only one such factor for a single boundary twist. The fibration base, precise boundary model, and comparison maps are absent. The source stresses that closed-surface equivalence is weaker, with larger equivalence classes, so that result does not settle the boundary-surface question posed here.

Short English statement

In the specified quotient of the six-strand braid group, classify positive factorizations of $\Delta_4^4$: must their lengths be exactly $40,30,20$ and their Hurwitz classes the three displayed ones? The closed genus-two fibration result is related context, with a weaker equivalence that need not answer the boundary case.

Sources

• Local UnsolvedMath registry, permanent ID 11000151, source code catalogue entry, unnumbered Question, PDF page 133. 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 volume contains 329 explicitly typeset headings across chapters, with repeated labels disambiguated by page.
• Reference [1] is cited for Hurwitz equivalence and the recent-to-the-source Siebert--Tian result. No full bibliographic entry, fibration-base hypotheses, or exact comparison theorem is provided.
• Supplied status open and OPEN-TRIAGE. The research summary says only "Truncated." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.