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 Haarrandom G -elements in the letters. This measure is determined by its Fourier coefficients: the w -expectations $${\mathbb E}_{w}[\chi]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:mo>[</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>]</mml:mo> </mml:math> of the irreducible characters of G . For every compact group G , the wreath product with the symmetric group G ≀ S n has some natural irreducible characters χ , and we approximate $${\mathbb E}_{w}[\chi]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:mo>[</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>]</mml:mo> </mml:math> for every word w ∈ F r , revealing new automorphism-invariant quantities of words that generalize the primitivity rank π ( w ). This generalizes previous works by Puder–Parzanchevski and Magee–Puder. We demonstrate applications to automorphism groups of trees, investigate properties of the new invariants, and show polynomial decay of $${\mathbb E}_{w}[\chi]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> </mml:mrow> </mml:msub> <mml:mo>[</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>]</mml:mo> </mml:math> also for wreath products with more general actions.