## Abstract

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^{∞} 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^{∞} 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 Γ, a random homomorphism from Γ to a finite simple group G is surjective with probability tending to 1 as |G|→∞.

Original language | American English |
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Pages (from-to) | 561-608 |

Number of pages | 48 |

Journal | Annals of Mathematics |

Volume | 190 |

Issue number | 2 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:Keywords: Waring problems, word maps, simple groups, uniform distributions, random walks, flat morphisms, one-relator groups AMS Classification: Primary: 20P05; Secondary: 11P05, 20C30, 20C33, 20D06, 20G40. ML was partially supported by NSF grants DMS-1401419 and DMS-1702152. AS was partially supported by ISF grant 686/17 and the Vinik Chair of mathematics, which he holds. PT was partially supported by NSF grant DMS-1840702 and the Joshua Barlaz Chair in Mathematics. The authors were also partially supported by BSF grant 2016072. The paper is partially based upon work supported by the NSF under grant DMS-1440140 while AS and PT were in residence at MSRI (Berkeley, CA), during the Spring 2018 semester. It is a pleasure to thank the Institute for the hospitality and support. The authors are grateful to the referee for careful reading and insightful comments that helped greatly improve the paper. ©c 2019 Department of Mathematics, Princeton University.

Publisher Copyright:

© 2019 Department of Mathematics, Princeton University.

## Keywords

- Atmorphisms
- One-relator groups
- Random walks
- Simple groups
- Uniform distributions
- Waring problems
- Word maps