## 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

Publisher Copyright:© 2019 Department of Mathematics, Princeton University.

## Keywords

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