Discrete groups acting on complex projective spaces e Seade 1 Jos´ 1 Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma de M´ exico. LMS Singularity Day, Liverpool, G.B., March 29, 2016 Seade Discrete groups acting on complex projective spaces
Section 1: The classical case n = 1 = Aut Hol CP 1 ∼ • Recall: PSL ( 2 , C ) ∼ = Conf + ( S 2 ) ∼ = Iso + ( H 3 ) ∼ = ∼ = { az + b cz + d , a , b , c , d ∈ C , ad − bc = 1 } . • A Lie group, diffeomorphic (as manifold) to R 3 × SO ( 3 ) . • Its discrete subgroups are called Kleinian groups. • These are either infinite or conjugate to a finite subgroup of SU ( 2 ) • If Γ ⊂ PSL ( 2 , C ) is finite, then V Γ := C 2 / Γ is a complex analytic surface with an isolated singularity. Very interesting classical singularities: • Du Val sings., Arnold’s modality 0, Rational double points, etc. • These are all hypersurface singularities Seade Discrete groups acting on complex projective spaces
For instance, if Γ is the binary icosahedral group, then V Γ is: { ( z 1 , z 2 , z 3 ) ∈ C 3 | z 2 1 + z 3 2 + z 5 3 = 0 } , and its link is Poincare’s homology 3-sphere. If Γ is the cyclic group of order r > 1, then V Γ is: { ( z 1 , z 2 , z 3 ) ∈ C 3 | z 2 1 + z 2 2 + z r 3 = 0 } , and its link is a lens space L ( r , 1 ) , obtained by glueing two solid Tori S 1 × D 2 by a homeomorphism of their boundaries that takes a parallel into a curve of type ( r , 1 ) . This all goes back, essentially, to F . Klein’s work in the 19th Century. Focus today on infinite discrete subgroups . Seade Discrete groups acting on complex projective spaces
Notice CP 1 is compact, so every Γ -orbit has accumulation points: Definition The limit set of Γ ⊂ PSL ( 2 , C ) in CP 1 is the set Λ of accumulation points of all orbits. Seade Discrete groups acting on complex projective spaces
This has remarkable properties. Some basic ones are: Theorem Λ is a closed invariant set. 1 If Λ has finite cardinality, then it consists of 1 or 2 points, 2 and the group is called elementary. If we assume from now on that the group is non-elementary. Then: The action on Λ is minimal (every orbit is dense) 3 Λ is the set of accumulation points of every orbit. 4 And furthermore, if we are interested in complex geometry, then: Seade Discrete groups acting on complex projective spaces
This has remarkable properties. Some basic ones are: Theorem Λ is a closed invariant set. 1 If Λ has finite cardinality, then it consists of 1 or 2 points, 2 and the group is called elementary. If we assume from now on that the group is non-elementary. Then: The action on Λ is minimal (every orbit is dense) 3 Λ is the set of accumulation points of every orbit. 4 And furthermore, if we are interested in complex geometry, then: Seade Discrete groups acting on complex projective spaces
This has remarkable properties. Some basic ones are: Theorem Λ is a closed invariant set. 1 If Λ has finite cardinality, then it consists of 1 or 2 points, 2 and the group is called elementary. If we assume from now on that the group is non-elementary. Then: The action on Λ is minimal (every orbit is dense) 3 Λ is the set of accumulation points of every orbit. 4 And furthermore, if we are interested in complex geometry, then: Seade Discrete groups acting on complex projective spaces
This has remarkable properties. Some basic ones are: Theorem Λ is a closed invariant set. 1 If Λ has finite cardinality, then it consists of 1 or 2 points, 2 and the group is called elementary. If we assume from now on that the group is non-elementary. Then: The action on Λ is minimal (every orbit is dense) 3 Λ is the set of accumulation points of every orbit. 4 And furthermore, if we are interested in complex geometry, then: Seade Discrete groups acting on complex projective spaces
This has remarkable properties. Some basic ones are: Theorem Λ is a closed invariant set. 1 If Λ has finite cardinality, then it consists of 1 or 2 points, 2 and the group is called elementary. If we assume from now on that the group is non-elementary. Then: The action on Λ is minimal (every orbit is dense) 3 Λ is the set of accumulation points of every orbit. 4 And furthermore, if we are interested in complex geometry, then: Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
Theorem Its complement Ω is the set where the action is 1 discontinuous. Ω actually is the maximal set where the action is properly 2 discontinuous. It is also the equicontinuity set of the action. If Ω has finitely many connected components, then it has at 3 most two (it can be empty). The quotient Ω / Γ is a Riemann surface with a projective 4 orbifold structure. Every Riemann surface can be realized in this way 5 (Koebe’s retrosection theorem) If Γ is finitely generated, then Ω / Γ is of finite type (Sullivan), 6 i.e. It has finitely many connected components, each being a compact Riemann surface minus finitely many points. Seade Discrete groups acting on complex projective spaces
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