Fibers of word maps and some applications

Michael Larsen*, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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| 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 languageAmerican English
Pages (from-to)36-48
Number of pages13
JournalJournal of Algebra
Issue number1
StatePublished - 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: (M. Larsen), (A. Shalev).


  • Finite simple groups
  • Simple algebraic groups
  • Word maps


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