Word maps have large image
Group Theory
2007-05-23 v2
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.
Cite
@article{arxiv.math/0211302,
title = {Word maps have large image},
author = {Michael Larsen},
journal= {arXiv preprint arXiv:math/0211302},
year = {2007}
}
Comments
9 pages; Theorem 1 credited to A. Borel with the relevant bibliographic entry supplied