Uniform bounded generation conjecture for principal congruence subgroups

From papers

Let RR be a ring of algebraic integers with infinitely many units, let II be a non-trivial ideal in RR, and let Φ\Phi be an irreducible root system of rank at least 22. Write NI,ΦN_{I,\Phi} for the principal congruence subgroup and

Q(Φ,I)={Aεϕ(x)A1xI, AG(Φ,R), ϕΦ}.Q(\Phi,I)=\{A\varepsilon_{\phi}(x)A^{-1}\mid x\in I,\ A\in G(\Phi,R),\ \phi\in\Phi\}.

Uniform bounded generation conjecture. There is a minimal constant K:=K(Φ)K:=K(\Phi) proportional to rank(Φ){\rm rank}(\Phi) and independent of RR and II such that

NI,Φ=Q(Φ,I)K.N_{I,\Phi}=Q(\Phi,I)^K.

This conjecture asks for a bound on bounded generation by a conjugacy-invariant collection that depends only on the root system, rather than on the ring or ideal. The supplied text gives no evidence that the conjecture has been resolved.

Progress summary

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Sources & referencesView supporting material

Primary source

Alexander Alois Trost, “Bounded generation for congruence subgroups of Sp_4(R)”, arXiv:2101.02301 (2021).

Additional references

2 papers in this index state this conjecture (2020–2021). The statement above is taken from the most recent of them; the others are arXiv:2012.12328.

Solutions 0

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