Word maps have large image

2002/11/30 by Michael Larsen · 1 citation
Mathematics · #math.GR #msc:20G15 #msc:20D06

paper · pdf

published as Israel J. Math. 139 (2004), 149--156 · 9 pages; Theorem 1 credited to A. Borel with the relevant bibliographic entry supplied

arxiv created 2004/04/20 · arxiv updated 2009/11/30

Abstract

An element w in the free group on r letters defines a map f from G^r 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 log |f(G^r)| over log |G| tends to 1 as the order of G tends to infinity.

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