Absolute profinite rigidity and hyperbolic geometry
Copyright policy )Absolute profinite rigidity and hyperbolic geometry
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[��])$ with $��^2+��+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 2020
- McMaster University Canada
- Purdue University System United States
- Purdue University West Lafayette United States
- Rice University United States
- RICE UNIVERSITY
Mathematics - Number Theory, Structure of modular groups and generalizations; arithmetic groups, Geometric Topology (math.GT), Group Theory (math.GR), Mathematics - Geometric Topology, Bianchi group, rigidity, hyperbolic 3-orbifold, General geometric structures on low-dimensional manifolds, FOS: Mathematics, Number Theory (math.NT), Limits, profinite groups, Fuchsian groups and their generalizations (group-theoretic aspects), profinite completion, Mathematics - Group Theory, hyperbolic 3-manifold, Weeks manifold
Mathematics - Number Theory, Structure of modular groups and generalizations; arithmetic groups, Geometric Topology (math.GT), Group Theory (math.GR), Mathematics - Geometric Topology, Bianchi group, rigidity, hyperbolic 3-orbifold, General geometric structures on low-dimensional manifolds, FOS: Mathematics, Number Theory (math.NT), Limits, profinite groups, Fuchsian groups and their generalizations (group-theoretic aspects), profinite completion, Mathematics - Group Theory, hyperbolic 3-manifold, Weeks manifold
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