Sparse Random Covers and Growth of Torsion in First Homology

2026/08/05 by Raz Slutsky
Mathematics · #math.GR #math.DG #math.MG #math.NT #msc:22E40 #msc:57M07 #msc:11F75 #msc:20F69 #msc:53C35

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19 pages

arxiv created 2026/08/05 · arxiv updated 2026/08/06

Abstract

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let X=G/K be a symmetric space of noncompact type and real rank r≥2. We prove that there are constants C_X,R_X<∞, depending only on X, such that every torsion-free lattice Γ<G, with M=Γ\backslash X and global injectivity radius R=InjRad(M)≥ R_X, satisfies max\d(Γ), log|H₁(M;\mathbb Z)_tors|\ ≤ C_Xvol(M)R^(1-r)/2(log R)² where d(Γ) denotes the minimal size of a generating set. We then prove the corresponding vanishing statements along Benjamini--Schramm convergent sequences. The vanishing of normalized torsion in first homology answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. Finally, we prove the analogous statements for affine buildings.

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