A word is an element in a free group. Given a word $w = w(x_1, \dots, w_k) \in F_d$ and a group G, we have the word map $w: G^k \to G$ defined by substitution. The set of values w(G) consists of the image of this map and the inverses of elements of the image. The width of the word w in the group G is the minimal constant C such that every element of <w(G)> can be expressed as the product of C elements of w(G). The talk will be devoted to known results about width of words in certain linear groups, such as algebraic groups over an algebraically closed field, compact Lie groups, finite simple groups, general linear groups over a skew field, and Chevalley groups over commutative rings. The following recent result by the speaker will be discussed in detail: Let \Phi be an irreducible root system of rank at least 2. For every positive integer d there exists a constant $C(\Phi, d)$ such that for every ring R which is a localization of the ring of integers of a number field of degree d (with certain additional assumption for the root systems C_2 and G_2) the width of any word in the simply connected Chevalley group $G(\Phi,R)$ is at most $C(\Phi,d)$.

# Cable Car Algebra Seminar: Width of words in linear groups.

Abstract: