Djoković’s Conjecture A for PU(n,1)

For every integer $n\ge 2$ , the involution length of $\mathrm{PU}(n,1)$ is exactly $4$ : every $g\in\mathrm{PU}(n,1)$ can be written as a product of at most four nontrivial involutions, and some element cannot be written as a product of three involutions. Moreover, every $g\in\mathrm{PU}(n,1)$ is a single commutator, i.e. there exist $a,b\in\mathrm{PU}(n,1)$ such that $g=aba^{-1}b^{-1}$ .

References

Primary source

arXiv

Additional references

Involution and Commutator Length in PU(n,1) — arXiv — Zhongqi Wang, Shihai Yang

complex-hyperbolic-geometry group-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjecture in every dimension, but the result has not yet been refereed.

Djoković’s Conjecture A concerns the exact involution length and single-commutator property of $\mathrm{PU}(n,1)$ for all $n$ . The latest preprint claims both statements for the full family.

Known results

$\mathrm{PU}(2,1)$ has involution length $4$ and commutator length $1$ ; for $\mathrm{PU}(n,1)$ with $n\geq 3$ , only the bound $\leq 8$ was previously known (Gongopadhyay and Thomas, 2016).

September 2026 preprint

Zhongqi Wang and Shihai Yang claim that the higher-dimensional involution-length bound improves to the optimal value $4$ and that every element of $\mathrm{PU}(n,1)$ is a single commutator. This would settle Conjecture A for all $n$ , but the preprint is unrefereed.

Current status (as of September 2026): A preprint claims the conjecture is solved for all $\mathrm{PU}(n,1)$ , extending the verified two-dimensional results, but its full proof remains unverified.

Sources

No solutions have been posted yet.