Fibers of word maps and some applications

Michael Larsen; Aner Shalev

doi:10.1016/j.jalgebra.2011.10.040

Fibers of word maps and some applications

Michael Larsen^*, Aner Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

24 Scopus citations

Abstract

Every word w in the free group F _d defines for each group G a word map, also denoted w, from G ^d to G. We prove that for all w≠1 there exists ε>0 such that for all finite simple groups G and all g∈G,|w-1(g)|=O(|G|d-ε), where the implicit constant depends only on w. In particular the probability that w(g1,..., gd)=1 is at most |G| ^-ε for some ε>0 and all large finite simple groups G. This result is then applied in the context of subgroup growth and representation varieties.

Original language	English
Pages (from-to)	36-48
Number of pages	13
Journal	Journal of Algebra
Volume	354
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2011.10.040
State	Published - 15 Mar 2012

Bibliographical note

Funding Information:
✩ Michael Larsen was partially supported by NSF Grant DMS-0800705. Aner Shalev was partially supported by ERC Advanced Grant 247034. Both authors were partially supported by Bi-National Science Foundation United States–Israel Grant 2008194. * Corresponding author. E-mail addresses: [email protected] (M. Larsen), [email protected] (A. Shalev).

Keywords

Finite simple groups
Simple algebraic groups
Word maps

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10.1016/j.jalgebra.2011.10.040

Cite this

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keywords = "Finite simple groups, Simple algebraic groups, Word maps",

author = "Michael Larsen and Aner Shalev",

note = "Funding Information: ✩ Michael Larsen was partially supported by NSF Grant DMS-0800705. Aner Shalev was partially supported by ERC Advanced Grant 247034. Both authors were partially supported by Bi-National Science Foundation United States–Israel Grant 2008194. * Corresponding author. E-mail addresses: [email protected] (M. Larsen), [email protected] (A. Shalev).",

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Fibers of word maps and some applications

Abstract

Bibliographical note

Keywords

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