Word Measures On Wreath Products I

Abstract

Every word $w$ in the free group $F_r$ of rank $r$ induces a probability measure (the $w$-measure) on every compact group $G$, by substitution of Haar-random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $\mathbb{E}_w[\chi]$ of the irreducible characters of $G$. For every compact group $G$, the wreath product with the symmetric group $G\wr S_n$ has some natural irreducible characters $\chi$, and we approximate $\mathbb{E}_w[\chi]$ for every word $w\in F_r$, revealing new automorphism-invariant quantities of words that generalize the primitivity rank $\pi(w)$. This generalizes previous works by Parzanchevsky-Puder and Magee-Puder. We demonstrate applications to automorphism groups of trees, investigate properties of the new invariants, and show polynomial decay of $\mathbb{E}_w[\chi]$ also for wreath products with more general actions.