Probabilistic Waring problems for finite simple groups

doi:10.4007/annals.2019.190.2.3

Probabilistic Waring problems for finite simple groups

2019-09-01

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DOI：10.4007/annals.2019.190.2.3

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The probabilistic Waring problem for finite simple groups asks whether every word of the form w(1)w(2) , where w(1) and w(2) are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the L-1 norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic L-infinity Waring problem for finite simple groups. We show that for every l >= 1, there exists (an explicit) N = N(l) = O(l(4)), such that if w(1) , ..., w(N) are non-trivial words of length at most l in pairwise disjoint sets of variables, then their product w(1 )... w(N) is almost uniform on finite simple groups with respect to the L-infinity norm. The dependence of N on l is genuine. This result implies that, for every word w = w(1 )... w(N) as above, the word map induced by w on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented. In particular, we show that, for certain one-relator groups Gamma, a random homomorphism from Gamma to a finite simple group G is surjective with probability tending to 1 as vertical bar G vertical bar -> infinity.

Keywords:

Waring problems

word maps

simple groups

uniform distributions

random walks

flat morphisms

one-relator groups

Probabilistic Waring problems for finite simple groups

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