A representation of Out(Fn) by counting subwords of cyclic words

We generalize the combinatorial approaches of Rapaport and Higgins–Lyndon to the Whitehead algorithm. We show that for every automorphism φ of a free group F and every word u∈F there exists a finite multiset of words Su,φ satisfying the following property: For every cyclic word w, the number of times u appears as a subword of φ(w) depends only on the appearances of words in Su,φ as subwords of w. We use this fact to construct a faithful representation of Out(Fn) on an inverse limit of Z-modules, so that each automorphism is represented by sequence of finite rectangular matrices, which can be seen as successively better approximations of the automorphism.