2024/02/16 by Nir Avni, Avni, Nir, Itay Glazer +3
Mathematics · #14B05 (Secondary) #20P05 (Primary) 60B15 #22E46 #43A25 #60B20 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2402.11108
openalex publication_date 2024/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To a non-trivial word w(x1,...,xr) in a free group Fr on r elements and a group G, one can associate the word map wG:Gr→ G that takes an r-tuple (g1,...,gr) in Gr to w(g1,...,gr). If G is compact, we further associate the word measure τw,G, defined as the distribution of wG(X1,...,Xr), where X1,...,Xr are independent and Haar-random elements in G. In this paper we study word maps and word measures on the family of special unitary groups \ SUn\ n≥2. Our first result is a small ball estimate for w_SUn. We show that for every w∈ Fr\smallsetminus\ 1\ there are ε(w),δ(w)>0 such that if B\subseteqSUn is a ball of radius at most δ(w)diam(SUn) in the Hilbert-Schmidt metric, then τ_w,SUn(B)≤(μ_SUn(B))ε(w), where μ_SUn is the Haar probability measure. Our second main result is about the random walks generated by τ_w,SUn. We provide exponential upper bounds on the large Fourier coefficients of τ_w,SUn, and as a consequence we show there exists t(w)∈ℕ, such that τ_w,SUn*t has bounded density for every t≥ t(w) and every n≥2, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character ρ of SUn, an exponential upper bound of the form |ρ(g)|<ρ(1)1-ε, for elements g in SUn whose eigenvalues are sufficiently spread out on the unit circle in \mathbbC×.