Math 2.0

Functions of a complex variable Definition

Kleinian group

Builds on Discrete subgroupSL(2,C)Möbius transformationRiemann sphere

Definition

Let PSL2(ℂ)=SL2(ℂ)/{±I}\mathrm{PSL}_2(\C) = \mathrm{SL}_2(\C) / \set{\pm I}, where SL(2,C) has the topology of a subset of ℂ4\C^4 and PSL2(ℂ)\mathrm{PSL}_2(\C) has the quotient topology. A Kleinian group is a discrete subgroup of PSL2(ℂ)\mathrm{PSL}_2(\C).

Remarks

The group PSL2(ℂ)\mathrm{PSL}_2(\C) acts on the Riemann sphere by Möbius transformations and on hyperbolic 33-space ℍ3\mathbb H^3 by isometries. A subgroup is Kleinian if and only if its action on ℍ3\mathbb H^3 is properly discontinuous. Some authors also require that the group act properly discontinuously on a nonempty open subset of the Riemann sphere.

Example: the Picard group PSL2(ℤ[i])\mathrm{PSL}_2(\Z[i]), of matrices with Gaussian integer entries, is Kleinian. Distinct Gaussian integers are at distance at least 11, so a small neighbourhood of ±I\pm I contains no other element of the group.

Non-example: the group PSL2(ℤ[2])\mathrm{PSL}_2(\Z[\sqrt2]) is not Kleinian. It contains the translations z↦z+a+b2z \mapsto z + a + b\sqrt2 with a,b∈ℤa, b \in \Z, and their translation lengths come arbitrarily close to 00 without being 00.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

Show the path as a graph
Cross product, concept, no evidence levelCross productconceptDeterminant, concept, no evidence levelDeterminantconceptDiscrete subgroup, concept, no evidence levelDiscrete subgroupconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptReal projective space, object, no evidence levelReal projective spaceobjectRiemann sphere, concept, no evidence levelRiemann sphereconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial linear group, object, no evidence levelSpecial linear groupobjectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)exampleSL(2,C), example, no evidence levelSL(2,C)exampleKleinian group, definition, no evidence levelKleinian groupdefinition
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. Cross product concept
  2. Determinant concept
  3. Discrete subgroup concept
  4. General linear group object
  5. Group concept
  6. Hermitian matrix concept
  7. Identity matrix object
  8. Lie group concept
  9. Matrix definition
  10. Möbius transformation concept
  11. Real projective space object
  12. Riemann sphere concept
  13. Vector space definition
  14. Ring definition
  15. Square matrix definition
  16. Transpose definition
  17. Linear map definition
  18. Unit of a ring definition
  19. Conjugate transpose definition
  20. Multilinear map definition
  21. Orthogonal matrix definition
  22. Division ring definition
  23. Alternating multilinear map definition
  24. Orthogonal group object
  25. Unitary matrix definition
  26. The quaternions example
  27. Determinant definition
  28. Unitary group object
  29. Special orthogonal group object
  30. Special linear group object
  31. Special unitary group SU(n) example
  32. Rotation group SO(3) example
  33. SU(2) example
  34. SL(2,C) example
  35. Kleinian group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kleinian_group
About this page
TagMC4EFHWRJG
URIm2://MC4EFHWRJG
Kinddefinition
WritingW0 stub
MSC30F40, 57K32, 20H10
DefinesKleinian group
UsesDiscrete subgroup, SL(2,C), Möbius transformation, Riemann sphere

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