## Abstract

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map (x, y) ↦ x^{N}y^{N} is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map (x, y, z) ↦ x^{N}y^{N}z^{N} is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map (x, y) ↦ x^{N}y^{N} that depend on the number of prime factors of the integer N.

Original language | American English |
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Pages (from-to) | 589-695 |

Number of pages | 107 |

Journal | Inventiones Mathematicae |

Volume | 213 |

Issue number | 2 |

DOIs | |

State | Published - 1 Aug 2018 |

### Bibliographical note

Funding Information:Acknowledgements The first author was partially supported by the NSF Grants DMS-1302886 and DMS-1600056, and the Simons Foundation Fellowship 224965. He also thanks the Institute for Advanced Study for its support. The third author was partially supported by the Marsden Fund of New Zealand via Grant UOA 1626. The fourth author was supported by ERC Advanced Grant 247034, ISF Grant 1117/13 and the Vinik Chair of Mathematics which he holds. The fifth author was partially supported by the NSF Grants DMS-1201374 and DMS-1665014, the Simons Foundation Fellowship 305247, the Mathematisches Forschungsinstitut Oberwolfach, and the EPSRC. Parts of the paper were written while the fifth author visited the Department of Mathematics, Harvard University, and Imperial College, London. He thanks Harvard University and Imperial College for generous hospitality and stimulating environments. The authors thank Frank Lübeck for providing them with the proof of Lemma 7.8.

Funding Information:

The first author was partially supported by the NSF Grants DMS-1302886 and DMS-1600056, and the Simons Foundation Fellowship 224965. He also thanks the Institute for Advanced Study for its support. The third author was partially supported by the Marsden Fund of New Zealand via Grant UOA 1626. The fourth author was supported by ERC Advanced Grant 247034, ISF Grant 1117/13 and the Vinik Chair of Mathematics which he holds. The fifth author was partially supported by the NSF Grants DMS-1201374 and DMS-1665014, the Simons Foundation Fellowship 305247, the Mathematisches Forschungsinstitut Oberwolfach, and the EPSRC. Parts of the paper were written while the fifth author visited the Department of Mathematics, Harvard University, and Imperial College, London. He thanks Harvard University and Imperial College for generous hospitality and stimulating environments. The authors thank Frank L?beck for providing them with the proof of Lemma 7.8. The authors are grateful to the referee for careful reading and insightful comments on the paper that helped improve both the exposition of the paper and some of its results.

Publisher Copyright:

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

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