Math 2.0

Topological groups, Lie groups Transformation groups and discrete subgroups of Lie groups Theorem

Wang's finiteness theorem

Builds on Lattice in a Lie groupSemisimple Lie groupHaar measureKazhdan–Margulis theoremWeil's local rigidity theorem

Statement

Let GG be a connected semisimple Lie group with finite center and without compact factors. Assume that no simple factor of GG is locally isomorphic to SL2(ℝ)\mathrm{SL}_2(\R) or to SL2(ℂ)\mathrm{SL}_2(\C). Fix a Haar measure on GG, and for a lattice Γ\Gamma of GG let covol⁡(Γ)\operatorname{covol}(\Gamma) be the measure of G/ΓG/\Gamma for the induced invariant measure. Then for every v>0v > 0 there are only finitely many lattices Γ\Gamma in GG with covol⁡(Γ)≤v\operatorname{covol}(\Gamma) \le v, up to conjugation by elements of GG.

Remarks

The result is due to H. C. Wang. The proof combines two facts. The Kazhdan–Margulis theorem gives a fixed neighbourhood of the identity that every lattice avoids up to conjugation, which makes the set of lattices of bounded covolume compact up to conjugation. Weil's local rigidity theorem and its extension to non-uniform lattices make each lattice isolated up to conjugation, and a compact set of isolated points is finite. The excluded factors are necessary. Fix a genus g≥2g \ge 2. In PSL2(ℝ)\mathrm{PSL}_2(\R) the fundamental groups of closed hyperbolic surfaces of genus gg all have covolume 4π(g−1)4\pi(g - 1), by the Gauss–Bonnet theorem. Yet they form uncountably many conjugacy classes, parametrised by Teichmüller space. In PSL2(ℂ)\mathrm{PSL}_2(\C) hyperbolic Dehn filling gives infinitely many closed hyperbolic 33-manifolds whose volumes are bounded by the volume of one cusped hyperbolic 33-manifold.

Going deeper

For a student

Wang's theorem is the global consequence of two local facts: lattices cannot contain elements too close to the identity in a uniform way, and lattices cannot be deformed. It fails for factors of type SL2(ℝ)\mathrm{SL}_2(\R), where lattices deform continuously. It also fails for SL2(ℂ)\mathrm{SL}_2(\C), where hyperbolic Dehn filling produces infinitely many lattices of bounded covolume. It is a starting point for counting lattices of bounded covolume, and for the study of the smallest covolume of a lattice in a given group.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Discrete subgroup, concept, no evidence levelDiscrete subgroupconceptGroup homomorphism, concept, no evidence levelGroup homomorphismconceptHaar measure, concept, no evidence levelHaar measureconceptLattice in a Lie group, concept, no evidence levelLattice in a Lie groupconceptNeighbourhood, concept, no evidence levelNeighbourhoodconceptSemisimple Lie group, concept, no evidence levelSemisimple Lie groupconceptKazhdan–Margulis theorem, statement, no evidence levelKazhdan–Margulis theo…statementWeil's local rigidity theorem, statement, no evidence levelWeil's local rigidity…statementWang's finiteness theorem, statement, no evidence levelWang's finiteness the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Discrete subgroup concept
  2. Group homomorphism concept
  3. Haar measure concept
  4. Lattice in a Lie group concept
  5. Neighbourhood concept
  6. Semisimple Lie group concept
  7. Kazhdan–Margulis theorem statement
  8. Weil's local rigidity theorem statement
  9. Wang's finiteness theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup), Rigidity results
About this page
TagSP421Y498C
URIm2://SP421Y498C
Kindstatement, theorem
WritingW0 stub
MSC22E40
StatusThis copy of the site has no build data. Run m2 status SP421Y498C.
AboutLattice in a Lie group
UsesLattice in a Lie group, Semisimple Lie group, Haar measure, Kazhdan–Margulis theorem, Weil's local rigidity theorem

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