Subword complexity is a basic invariant for words on a finite alphabet. I will explain how one can define a complexity for points in the boundary of a finitely generated free group F or for a lamination on F. This complexity, or rather the way it grows, is invariant under automorphisms of F and may be interpreted geometrically. I will discuss a version of Pansiot's theorem about the complexity of fixed points of substitutions in the context of automorphisms of free groups. This is based on joint work with Arnaud Hilion.
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