SL2-factorizations of Chevalley groups

Abstract

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.