A note on maximal lattice growth in SO(1,n)

TL;DR

This paper demonstrates that the number of noncommensurable and maximal lattices in SO(1,n) grows at least exponentially by constructing large families of hybrid hyperbolic manifolds.

Contribution

It introduces a method to construct large families of noncommensurable hybrid hyperbolic manifolds, establishing exponential growth in the count of such lattices.

Findings

Number of noncommensurable lattices in SO(1,n) is at least exponential.

Constructs large families of hybrid hyperbolic manifolds.

Shows existence of many maximal lattices in SO(1,n).

Abstract

We show that the number of noncommensurable lattices, hence also that of maximal lattices in SO(1,n) is at least exponential. To do so we construct large families of noncommensurable hybrid hyperbolic (Gromov/Piatetski-Shapiro) manifolds.