Publication Date:
2014-07-17
Description:
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.
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
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