For Gromov-Piatetski-Shapiro hybrid hyperbolic n-manifolds, the number of codimension-one totally geodesic hyperplanes is bounded by an explicit (if weak) function of volume and dynamical constants.
Arithmeticity, superrigidity, and totally geodesic submanifolds
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Quantitative finiteness of hyperplanes in hybrid manifolds
For Gromov-Piatetski-Shapiro hybrid hyperbolic n-manifolds, the number of codimension-one totally geodesic hyperplanes is bounded by an explicit (if weak) function of volume and dynamical constants.