Growth of primitive elements in free groups
Doron Puder, Conan Wu · 2016
Let pk,N be the number of primitive words of length N in the free group Fk, k ≥ 3. We show that the exponential growth rate of pk,N is 2k − 3, answering a question from the list ‘Open problems in combinatorial group theory ’ [Baumslag-Myasnikov-Shpilrain 02’]. Moreover, we show that a generic conjugacy class of primitive elements has a representative containing some letter exactly once. Our proof also works for giving the exact growth rate of the larger class of elements belonging to a proper free factor. 1