Quantitative finiteness of hyperplanes in hybrid manifolds

2025/06/17 by K. W. Ohm, Anthony Sanchez, Ohm, Ko W. +1
Computer Science · Mathematics · #22E40 (Secondary) #22F30 (Primary) #37D40 #Computational Geometry and Mesh Generation #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2506.14478

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We prove a quantitative finiteness theorem for the number of totally geodesic hyperplanes of non-arithmetic hyperbolic n-manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro for n≥3. This extends work of Lindenstrauss-Mohammadi in dimension 3. This follows from effective density theorem for periodic orbits of SO(n-1,1) acting on quotients of SO(n,1) by a lattice for n≥3. The effective density result uses a number of a ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures on the horospherical subgroup that are nearly full dimensional.

Citations

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Related

• A note on maximal lattice growth in SO(1,n)
• Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds
• Arithmeticity, Superrigidity, and Totally Geodesic Submanifolds
• Efficient fundamental cycles of cusped hyperbolic manifolds
• Counting commensurability classes of hyperbolic manifolds