Growth of primitive elements in free groups

D. Puder, C. Wu

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


In the free group F-{k}, an element is said to be primitive if it belongs to a free generating set. In this paper, we describe what a generic primitive element looks like. We prove that up to conjugation, a random primitive word of length N contains one of the letters exactly once asymptotically almost surely (as N\to \infty). This also solves a question raised by Baumslag-Myasnikov- Shpilrain (Contemp. Math. 296 (2002) 1-38). Let p-{k,N} be the number of primitive words of length N in F-{k}. We show that for k\geq 3, the exponential growth rate of p-{k,N} is 2k-3. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor.

Original languageEnglish
Pages (from-to)89-104
Number of pages16
JournalJournal of the London Mathematical Society
Issue number1
StatePublished - Aug 2014
Externally publishedYes


FundersFunder number
Seventh Framework Programme226135


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