Probabilistic Waring problems for finite simple groups

Michael Larsen, Aner Shalev, Pham Huu Tiep

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

The probabilistic Waring problem for finite simple groups asks whether every word of the form w1w2, where w1 and w2 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(l4), such that if w1,...,wN are non-trivial words of length at most l in pairwise disjoint sets of variables, then their product w1...wN 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=w1...wN 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 languageEnglish (US)
Pages (from-to)561-608
Number of pages48
JournalAnnals of Mathematics
Volume190
Issue number2
DOIs
StatePublished - Jan 1 2019

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'Probabilistic Waring problems for finite simple groups'. Together they form a unique fingerprint.

  • Cite this