Width of SL(n,𝒪_S,I)

Bounded generation for congruence subgroups of Sp4(R)

“…Remark 3.2. The A 2 -case for number fields is almost directly taken from [11,Corollary 3.3] and the result for the other roots system also uses the strategy from [11]. However, one could avoid [11] entirely and replace it with results from [18,25].…”

“…The A 2 -case for number fields is almost directly taken from [11,Corollary 3.3] and the result for the other roots system also uses the strategy from [11]. However, one could avoid [11] entirely and replace it with results from [18,25]. This approach could still be used to prove Theorem 1.2, but it would be completely unclear how large k has to be chosen beyond the vague fact that it has to be chosen larger than some threshold determined by an unknown function admitting only the root system Φ and the numbers [K : Q] (in the number field case) and [K : F(T )], |F| (in the global function field case) as arguments.…”

“…This approach could still be used to prove Theorem 1.2, but it would be completely unclear how large k has to be chosen beyond the vague fact that it has to be chosen larger than some threshold determined by an unknown function admitting only the root system Φ and the numbers [K : Q] (in the number field case) and [K : F(T )], |F| (in the global function field case) as arguments. Using the approach from [11] produces explicit thresholds (at least in most cases) and makes it clear which quantities have to be determined to make non-explicit threshold functions explicit.…”

“…The paper [11] only contains a special version of this fact in case of Φ = A 2 , but the proof works in general. This allows to deduce from bounded generation by root elements for Chevalley groups over R bounded generation by conjugates of root elements for the corresponding congruence subgroup C(Φ, I).…”

“…Citing results from [9] and [21], we will note in Subsection 3.1.3, that the ring R also has stable range at most 2 and so using the standard stable range arguments (as for example in the proof of [8,Main Theorem]), one can reduce any element of SL 3 ( R) to an element of its subgroup SL 2 ( R) by multiplying with 7 elementary matrices. The proof then proceeds by applying [11,Proposition 4.1] and the proof is generally very similar to the one in [11].…”