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Words and mixing times in finite simple groups. (English) Zbl 1245.20075

Let \(w=w(x_1,\dots,x_d)\) be a nontrivial group word, and for any group \(G\) write \(w(G)\) for the set of values of \(w\) on \(G\). It is known [M. Larsen, Isr. J. Math. 139, 149-156 (2004; Zbl 1130.20310)] that \(\log|w(G)|/\log|G|\to 1\) as \(G\) runs over any sequence of distinct (nonabelian) finite simple groups, and also that \(w(G)^3=G\) for all but a finite number of finite simple groups [see A. Shalev, Ann. Math. (2) 170, No. 3, 1383-1416 (2009; Zbl 1203.20013)].
For any nonempty subset \(W\) of \(G\) let \(P_W\) denote the probability distribution on \(G\) which is equal to \(1/|W|\) on \(W\) and \(0\) elsewhere. Let \(P_W*P_W\) denote the convolution of \(P_W\) with itself, \(U_G\) (\(=P_G\)) be the uniform distribution on \(G\), and \(\|\;\|_1\) be the \(1\)-norm on \(G\).
The main theorem of the present paper is that as \(G\) runs over the finite simple groups we have \(\| P_{w(G)}*P_{w(G)}-U_G\|_1\to 0\) as \(|G|\to\infty\). In particular, using this result, it can be proved that \(w(G)^2=G\) for all but a finite number of finite simple groups.

MSC:

20P05 Probabilistic methods in group theory
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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