N ov 2 00 2 WORD MAPS HAVE LARGE IMAGE
نویسنده
چکیده
An element w in the free group on r letters defines a map fw,G: G r → G for each group G. In this note, we show that whenever w 6= 1 and G is a semisimple algebraic group, fw,G is dominant. As an application, we show that for fixed w and Γi a sequence of pairwise non-isomorphic finite simple groups, lim i→∞ log |Γi| log |f w,Γi (Γ r i )| = 1. Let Fr be the free group on r generators x1, . . . , xr. For any group G, each word w = x1 a1x b2 a2 · · ·x bm am ∈ Fr defines a corresponding word map fw,G: G r → G: fw,G(g1, . . . , gr) = g b1 a1 g2 a2 · · · g bm am . The main result of this note is as follows: Theorem 1: If G is a simple algebraic group over any field K and w 6= 1, then fw,G is a dominant morphism. As an application, we prove the following theorem, which answers a question of A. Shalev: * Partially supported by NSF Grant DMS-0100537
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