Quantitative finiteness of hyperplanes in hybrid manifolds

TL;DR

This paper establishes a quantitative bound on the number of totally geodesic hyperplanes in certain non-arithmetic hyperbolic manifolds, extending previous results in dimension 3 using advanced dynamical and geometric techniques.

Contribution

It provides a new finiteness theorem for hyperplanes in higher-dimensional hyperbolic manifolds, generalizing prior work from dimension 3 to n≥3 with effective density estimates.

Findings

Bound on the number of hyperplanes in non-arithmetic hyperbolic manifolds

Extension of finiteness results to higher dimensions (n≥3)

Development of effective density and equidistribution techniques

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 $n\ge3$. This extends work of Lindenstrauss-Mohammadi in dimension 3. This follows from effective density theorem for periodic orbits of $\mathrm{SO}(n-1,1)$ acting on quotients of $\mathrm{SO}(n,1)$ by a lattice for $n\ge3$. 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.