Bounded Elementary Generation: There and back again
Abstract: It is known for some 2500 years, that an element of SL (2,Z) is a product of elementary transformation, but not a bounded such product. However, it is a product of not more than 37 such elementary transformations in SL (3,Z). There is no hope to generalise such an answer to arbitrary commutative rings, the elementary width is not bounded, in general. However, there are remarkable positive results for some special rings, and even more surprising such results that hold over arbitrary rings for some classes of matrices, such as commutators, etc. We will survey such results, and their generalisations to other groups, the relative case, etc. Even more amazingly, there are recent general results in the opposite direction, generalising the celebrated Brenner's lemma, namely bounded expressions of elementary transformations in terms of the elementary conjugates of an arbitrary matrix. We sketch possible applications to general word maps, etc.