Normalizers of lattices and isometry groups of arithmetic hyperbolic manifolds

2026/07/31 by Mikhail Belolipetsky, Tam Cheetham-West
Mathematics · #math.GT #math.GR #msc:57K32 #msc:57M50 #msc:57N16 #msc:22E40

paper · pdf

12 pages. Comments welcome

arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

We prove that every arithmetic lattice in PSL(2,ℂ) and every arithmetic lattice of the simplest type in PO(n,1), n≥ 2, is the normalizer of arbitrarily many of its sublattices. Combined with previous work, this result implies that every lattice in PSL(2,ℂ) has this property. In this way, we prove that the set of profinitely flexible lattices in PSL(2,ℂ) is either empty or countably infinite. Another result is that every finite group is realized as the full isometry group of an arithmetic hyperbolic n-manifold. The proof of this theorem is based on study of normalizers of lattices and subgroup growth theory.

Citations

• Finite groups and complex projective surfaces
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• Imitator homomorphisms for special cube complexes
• Subspace stabilisers in hyperbolic lattices
• Hierarchies for Relatively Hyperbolic Virtually Special Groups
• Peripheral fillings of relatively hyperbolic groups
• Counting maximal arithmetic subgroups
• Strong rigidity ofQ-rank 1 lattices