Probabilistic Waring problems for finite simple groups

Abstract

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, 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 $L^{\infty }$ Waring problem for finite simple groups. We show that for every $l \ge 1$, there exists (an explicit) $N = N(l)=O(l^4)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty }$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ 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 $|G| \to \infty $.

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      mrreviewer = {Jonathan Kujawa},
      doi = {10.1007/s00222-008-0145-7},
      url = {https://doi.org/10.1007/s00222-008-0145-7},
      zblnumber = {1166.20009},
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    @ARTICLE{LiSh2004,
      author = {Liebeck, Martin W. and Shalev, Aner},
      title = {Fuchsian groups, coverings of {R}iemann surfaces, subgroup growth, random quotients and random walks},
      journal = {J. Algebra},
      fjournal = {Journal of Algebra},
      volume = {276},
      year = {2004},
      number = {2},
      pages = {552--601},
      issn = {0021-8693},
      mrclass = {20H10 (20C30 20E07 20P05 28C10 57M07 60B15 60G50)},
      mrnumber = {2058457},
      mrreviewer = {Goulnara N. Arzhantseva},
      doi = {10.1016/S0021-8693(03)00515-5},
      url = {https://doi.org/10.1016/S0021-8693(03)00515-5},
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      title = {Fibers of word maps and some applications},
      journal = {J. Algebra},
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      volume = {354},
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      pages = {36--48},
      issn = {0021-8693},
      mrclass = {20D05 (20P05)},
      mrnumber = {2879221},
      mrreviewer = {Colva M. Roney-Dougal},
      doi = {10.1016/j.jalgebra.2011.10.040},
      url = {https://doi.org/10.1016/j.jalgebra.2011.10.040},
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      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {368},
      year = {2016},
      number = {3},
      pages = {1647--1661},
      issn = {0002-9947},
      mrclass = {20P05 (20C33 20D06)},
      mrnumber = {3449221},
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      doi = {10.1090/tran/6389},
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    @ARTICLE{LST,
      author = {Larsen, Michael and Shalev, Aner and Tiep, Pham Huu},
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      mrclass = {20D05 (11P05 20C33 20E05)},
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      author = {Liebeck, Martin W. and Shalev, Aner},
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      JOURNAL = {Proc. London Math. Soc. (3)},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {90},
      YEAR = {2005},
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      PAGES = {61--86},
      ISSN = {0024-6115},
      MRCLASS = {20C33 (20P05 60B15 60G50)},
      MRNUMBER = {2107038},
      DOI = {10.1112/S0024611504014935},
      URL = {https://doi.org/10.1112/S0024611504014935},
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      issn = {0020-9910},
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      author = {Liebeck, Martin W. and Shalev, Aner},
      title = {Character degrees and random walks in finite groups of {L}ie type},
      journal = {Proc. London Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {90},
      year = {2005},
      number = {1},
      pages = {61--86},
      issn = {0024-6115},
      mrclass = {20C33 (20P05 60B15 60G50)},
      mrnumber = {2107038},
      doi = {10.1112/S0024611504014935},
      url = {https://doi.org/10.1112/S0024611504014935},
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      author = {Shalev, Aner},
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      }

Authors

Michael Larsen

Indiana University, Bloomington, IN USA

Aner Shalev

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel

Pham Huu Tiep

Rutgers University, Piscataway, NJ USA