Products of Commutators on a General Linear Group Over a Division Algebra

Commutator Lengths in General Linear Group Over a Skew-Field

“…In this section we prove the theorem below; the proof is a modification of the proof of the main theorem in [4].…”

“…Since the reduced norms of the a i , and hence of the b i a i b −1 i , are pairwise distinct, it follows that h 1 and h 2 are D-regular (see Definition 3.3 and Proposition 3.4 in [4]). By [4,Proposition 4…”

“…By Theorem 2.1 in [4], which we referred to in the proof of Lemma 10, we may assume that g = vhu, where v ∈ V , u ∈ U , and…”

“…The skew-field was assumed to be finite-dimensional over its center and not coinciding with it. Later Kursov's result was rediscovered by Gordeev and Egorchenkova in [4] (see also [3]), where the corresponding analog of the Gauss decomposition with given semisimple part was proved.…”

“…However the main result of [5] and [4] is not optimal. As we show, for skew-fields that are finite dimensional over its center the following statement holds: if every element of the derived subgroup of the multiplicative group is a product of at most c commutators, then every element of the elementary subgroup is a product of at most c n commutators in the general linear group of n × n matrices, and also a product of at most c n−2 commutators in the elementary group, where x is the ceil function of x (Theorem 2).…”