%0 Journal Article
%T Growth of Primitive Elements in Free Groups
%A Doron Puder
%A Conan Wu
%J Mathematics
%D 2013
%I arXiv
%R 10.1112/jlms/jdu009
%X 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 from the list `Open problems in combinatorial group theory' [Baumslag-Myasnikov-Shpilrain 02']. Let $p_{k,N}$ be the number of primitive words of length $N$ in $F_k$. We show that for $k \ge 3$, 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.
%U http://arxiv.org/abs/1304.7979v3