Measure doubling in unimodular locally compact groups and quotients

2024/11/26 by Zuxiang Kong, Fei Peng, Kong, Zuxiang +3
Mathematics · #03C99 #11B30 (Secondary) #22D05 (Primary) 51F99 #22E30 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2411.17246

openalex publication_date 2024/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a (possibly discrete) unimodular locally compact group G with Haar measure μ_G, and a compact A⊆ G of positive measure with μ_G(A²)≤ Kμ_G(A). Let H be a closed normal subgroup of G and π: G → G/H be the quotient map. With the further assumption that A= A^-1, we show μ_G/H(πA ²) ≤ K² μ_G/H(πA). We also demonstrate that K² cannot be replaced by (1-ε)K² for any ε>0. In the general case (without A=A^-1), we show μ_G/H(πA ²) ≤ K³ μ_G/H(πA), improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set B⊆ A with μ_G(B)> μ_G(A)/2 such that μ_G/H(πB²) < 2K μ_G/H(πB).

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