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A note on maximal lattice growth in SO(1,\(n\)). (English) Zbl 1367.22005

For each \(n \geq 3\), it is established that the number of maximal lattices in \(\mathrm{SO}(n,1)\) with covolume less than \(v\) grows at least exponentially in \(v\). To do so, for every large \(m\) the author exhibits a family \(\{M_k\}\) of at least \(D^m\) pairwise non-commensurable closed hyperbolic \(n\)-manifolds of volume \(v \leq m\) where the constant \(D > 1\) depends on \(n\) only. These manifolds define \(D^m\) different commensurability classes of lattices in \(\mathrm{SO}(n,1)\) and picking a maximal group of minimal covolume in each class gives the result.
The manifolds \(M_k\) are constructed similarly as the famous non-arithmetic hybrids of M. Gromov and I. I. Piatetski-Shapiro [Publ. Math., Inst. Hautes Étud. Sci. 66, 93–103 (1988; Zbl 0649.22007)], except that one glues more than two arithmetic manifolds. One starts with a finite ordered family of pairwise non-commensurable, arithmetic hyperbolic \(n\)-manifolds which contain a fixed embedded nonseparating totally geodesic hypersurface \(\Sigma \). After deleting \(\Sigma\), these manifolds are completed to manifolds with boundary \(\Sigma \sqcup \Sigma\). These are glued in cyclic manner, according to the given order. The main point is that two so obtained manifolds are only commensurable if the two families have cyclically (or reflected cyclically) permuted order.
The result is to be contrasted with work of M. Belolipetsky [Duke Math. J. 140, No. 1, 1–33 (2007; Zbl 1131.22008)], who shows that the growth of maximal arithmetic lattices is subexponential. Also note that meanwhile, T. Gelander and A. Levit [Geom. Funct. Anal. 24, No. 5, 1431–1447 (2014; Zbl 1366.57011)] have identified the growth rate of the number of commensurability classes of hyperbolic \(n\)-manifolds with volume \(< v\) sharply: the number lies between \(v^{av}\) and \(v^{bv}\) for constants \(0 < a < b\).
As an aside the author constructs a sequence of hyperbolic \(n\)- manifolds with volume tending to infinity whose isometry groups have uniformly bounded order.

MSC:

22E40 Discrete subgroups of Lie groups
20E07 Subgroup theorems; subgroup growth
11E57 Classical groups
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