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  1. ; (, arXiv)
    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 LindenstraussMohammadi 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. 
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    Free, publicly-accessible full text available June 17, 2026