Word maps have large image

الأصل · EN

An element w in the free group on r letters defines a map f from Gʳ to G for each group G. In this note, we show that whenever w is non-trivial and G is a semisimple algebraic group, f is dominant. When G is a finite simple group, the ratio |f(Gʳ)| over |G| tends to 1 as the order of G tends to infinity.