Word maps have large image
Copyright policy )Word maps have large image
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.
9 pages; Theorem 1 credited to A. Borel with the relevant bibliographic entry supplied
- Indiana University United States
- Indiana University Bloomington United States
- DePaul University United States
- Indiana University Bloomington United States
20G15, word maps, dominant morphisms, 20D06, Group Theory (math.GR), Linear algebraic groups over arbitrary fields, finite simple groups, FOS: Mathematics, semisimple algebraic groups, Asymptotic results on counting functions for algebraic and topological structures, Mathematics - Group Theory, 20G15; 20D06
20G15, word maps, dominant morphisms, 20D06, Group Theory (math.GR), Linear algebraic groups over arbitrary fields, finite simple groups, FOS: Mathematics, semisimple algebraic groups, Asymptotic results on counting functions for algebraic and topological structures, Mathematics - Group Theory, 20G15; 20D06
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