Let m≥ 3, and let mathbfB_2,m(P^2) denote the relevant two-strand braid group on the projective plane with parameter m. Its lower central series (LCS) is the sequence of iterated commutator subgroups. Non-termination conjecture. The LCS of mathbfB_2,m(P^2) does not stop. The preceding computations show that, for all m≤ 1024 and for all m=2^nu with nu≤ 23, the LCS does not stop before the 100th term. The conjecture asserts that this behavior continues indefinitely for every m≥ 3.

The non-termination conjecture for the lower central series of two-strand braid groups on the projective plane

From papers

Let $m\geq 3$ , and let ${\mathbf{B}}_{2,m}({\mathbb{P}^2})$ denote the relevant two-strand braid group on the projective plane with parameter $m$ . Its lower central series (LCS) is the sequence of iterated commutator subgroups.

Non-termination conjecture. The LCS of ${\mathbf{B}}_{2,m}({\mathbb{P}^2})$ does not stop.

The preceding computations show that, for all $m\leq 1024$ and for all $m=2^\nu$ with $\nu\leq 23$ , the LCS does not stop before the 100th term. The conjecture asserts that this behavior continues indefinitely for every $m\geq 3$ .

Progress summary

Open

The conjecture remains open: extensive computations support it, but no proof or counterexample has been found.

The conjecture asserts that the lower central series of $\mathbf{B}_{2,m}(\mathbb{P}^{2})$ never terminates for every $m\geq 3$ . A 2022 study isolates this family as its sole unresolved case among the families examined.

Known results

Computations verify lower central series length at least $100$ for every $m\leq 1024$ .
Computations also verify length at least $100$ for $m=2^{\nu}$ with $\nu\leq 23$ .
These are finite computations and do not establish non-termination for all $m$ or indefinitely many terms.

2022 unresolved-case classification

The comprehensive study labels the assertion Conjecture 6.94 and explicitly leaves $\mathbf{B}_{2,m}(\mathbb{P}^{2})$ , $m\geq 3$ , unresolved. The scan found no subsequent proof, counterexample, claimed solution, verification, gap report, or retraction concerning this exact conjecture.

Current status (as of August 2026): The finite computational checks are established, but non-termination for every $m\geq 3$ remains unproved, with no known counterexample.

Sources

Sources & referencesView supporting material

Primary source

Jacques Darné, Martin Palmer and Arthur Soulié, “When the lower central series stops: a comprehensive study for braid groups and their relatives”, arXiv:2201.03542 (2022).

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