Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

• 成果类型: Article
• 署名作者: Fraczyk, Mikolaj
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-01021-1
• 发表日期: 2021
• 页码: 917-985
• 关键词: representations; formulas; number; regulator; RIGIDITY; GROWTH

摘要

We show that every sequence of torsion-free arithmetic congruence lattices in PGL(2, R) or PGL(2, 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.