On representing words in the automorphism group of the random graph. (English summary) J. Group Theory 9 (2006), no. 6, 815–836. Summary: “We discuss the solubility of equations of the form w = g, where w is a word (an element of a free group FX) and g is an element of a given group G. A word for which this equation is soluble for every g ∈ G is said to be universal for G. It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [R. L. Dougherty and J. Mycielski, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2233–2243; MR1605952 (99j:20002); R. C. Lyndon, in Mots, 143–152, Hermès, Paris, 1990; MR1252660 (95c:20006); J. Mycielski, Proc. Amer. Math. Soc. 100 (1987), no. 2, 237–241; MR0884459 (88c:20044)], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic ’ automorphisms, and the elements which suitably approximate them, called ‘special’.
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