The Waring Problem for Lie Groups and Chevalley Groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w 6= 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed word w 6= 1 and for a classical connected real compact Lie group G of sufficiently large rank we have w(G) = G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares. The first author was supported by ERC Advanced Grant no. 247034. The second author was partially supported by the Simons Foundation, the MSRI, NSF Grant DMS-1101424, and BSF Grant no. 2008194. The third author was partially supported by ERC Advanced Grant no. 247034, ISF grant no. 1117/13, BSF Grant no. 2008194 and the Vinik Chair of Mathematics which he holds. 1 2 CHUN YIN HUI, MICHAEL LARSEN, ANER SHALEV