Word maps have large image

An element omega in the free group on nu letters defines a map f(omega).(G):G(r) --> G for each group G. In this note we show that whenever omega not equal 1 and G is a semisimple algebraic group f(omega.G) is dominant. As an application, we show that for fixed omega and Gamma(i) a sequence of pairwise non-isomorphic finite simple groups. lim/i --> infinity log\Gamma(i)\/log\f(omega.Gammai)(Gamma(i)(r))\ = 1. Let F-r be the free group on r generators x(1)......x(r). For any group G. each word omega = x(a1)(b1)x(a2)(b2)...x(am)(bm) is an element of F-r defines a corresponding word map f(omega.G):G(r) --> G: f(omega.G)(g1.....gr) = g(a1)(b1)g(a2)(b2)...g(am)(bm).