2016/12/31 by Mikołaj Frączyk, Mikolaj Fraczyk
Mathematics · #Algebraic Geometry and Number Theory #Combinatorics #Congruence (geometry) #Conjecture #Geometric and Algebraic Topology #Geometry #Homotopy #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Multiplicity (mathematics) #Pure mathematics #Riemann hypothesis #Selberg trace formula #Torsion (gastropod) #math.GR #math.NT #msc:11F06 #msc:22E55
paper · pdf · doi:10.1007/s00222-020-01021-1
published as Frączyk, M. Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds. Invent. math. (2020) · 41 pages, condensed version rewritten according to referees' suggestions. Minor improvements in the main result
openalex publication_date 2020/11/19 · arxiv created 2020/11/20 · arxiv updated 2020/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that every sequence of torsion-free arithmetic congruence lattices in PGL(2,\mathbb R) or PGL(2,\mathbb C) satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for R>0 in certain range, growing linearly in the degree of the invariant trace field, the volume of the R-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3-manifold M is of order at most Vol(M)11/12. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic 3-manifolds: We show that there are constants A,B such that every such manifold M is homotopy equivalent to a simplicial complex with at most AVol(M) vertices, all of degrees bounded by B.