SL2(R) Dynamics and Harmonic Analysis Livio Flaminio [email protected] Institut de Mathématiques Université de Lille Nantes, 2-7 juillet 2017 R { g 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)={I g has trivial centre. Remarquable one-parameter subgroups n [ ] o n [ ] o n o cos θ − sin θ a 0 1 u K = ; A = −1 : a > 0 ; N = [ 0 1 ] ; sin θ cos θ θ2R 0 a u2R n [ ] o n o ? cosh θ sinh θ 1 0 A = sinh θ cosh θ ; N = [ v 1 ] ; θ2R v2R Introduction to SL2(R) Topology via remarkable actions The group SL2(R) and some subgroups Definition The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b 2 R − SL2( ) = c d : a; b; c; d ; ad bc = 1 : L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 Remarquable one-parameter subgroups n [ ] o n [ ] o n o cos θ − sin θ a 0 1 u K = ; A = −1 : a > 0 ; N = [ 0 1 ] ; sin θ cos θ θ2R 0 a u2R n [ ] o n o ? cosh θ sinh θ 1 0 A = sinh θ cosh θ ; N = [ v 1 ] ; θ2R v2R Introduction to SL2(R) Topology via remarkable actions The group SL2(R) and some subgroups Definition The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b 2 R − SL2( ) = c d : a; b; c; d ; ad bc = 1 : R { g 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)={I g has trivial centre. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 Introduction to SL2(R) Topology via remarkable actions The group SL2(R) and some subgroups Definition The group SL2(R) is the group of 2 × 2 real matrices of determinant 1: {[ ] } R a b 2 R − SL2( ) = c d : a; b; c; d ; ad bc = 1 : R { g 1 0 The largest normal subgroup of SL2( ) is its centre Z = I , I = [ 0 1 ]. Hence the quotient group PSL2(R) := SL2(R)={I g has trivial centre. Remarquable one-parameter subgroups n [ ] o n [ ] o n o cos θ − sin θ a 0 1 u K = ; A = −1 : a > 0 ; N = [ 0 1 ] ; sin θ cos θ θ2R 0 a u2R n [ ] o n o ? cosh θ sinh θ 1 0 A = sinh θ cosh θ ; N = [ v 1 ] ; θ2R v2R L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 2 33 • not compact (e.g. since N ≈ R is properly embedded); • connected; • not simply connected: in fact π1(G) = Z; • a simple Lie group. Properties SL2(R) is Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • not compact (e.g. since N ≈ R is properly embedded); • connected; • not simply connected: in fact π1(G) = Z; • a simple Lie group. Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • connected; • not simply connected: in fact π1(G) = Z; • a simple Lie group. Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • not simply connected: in fact π1(G) = Z; • a simple Lie group. Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected; L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 • a simple Lie group. Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected; • not simply connected: in fact π1(G) = Z; L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected; • not simply connected: in fact π1(G) = Z; • a simple Lie group. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) 4 The group SL2(R) as a sub-manifold of R inherits the differentiable structure and topology from R4. Properties SL2(R) is • not compact (e.g. since N ≈ R is properly embedded); • connected; • not simply connected: in fact π1(G) = Z; • a simple Lie group. Next we introduce some fundamental actions of SL2(R) which will be further exploited later on. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 3 33 Consequence As R2 n f0g and N are connected, G is connected. Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2. The orbits are: f0g and R2 n f0g. The stabiliser of the point (1; 0) is the group N. Hence we have 2 R n f0g = G:(1; 0) ≈ G= StabG [(1; 0)] = G=N: Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) Connectness G acts linearly on R2: [ ] a b c d :(x; y) = (ax + by; cx + dy) L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Consequence As R2 n f0g and N are connected, G is connected. Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) Connectness G acts linearly on R2: [ ] a b c d :(x; y) = (ax + by; cx + dy) The orbits are: f0g and R2 n f0g. The stabiliser of the point (1; 0) is the group N. Hence we have 2 R n f0g = G:(1; 0) ≈ G= StabG [(1; 0)] = G=N: L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2. Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) Connectness G acts linearly on R2: [ ] a b c d :(x; y) = (ax + by; cx + dy) The orbits are: f0g and R2 n f0g. The stabiliser of the point (1; 0) is the group N. Hence we have 2 R n f0g = G:(1; 0) ≈ G= StabG [(1; 0)] = G=N: Consequence As R2 n f0g and N are connected, G is connected. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Introduction to SL2(R) Topology via remarkable actions Topology of G = SL2(R) Connectness G acts linearly on R2: [ ] a b c d :(x; y) = (ax + by; cx + dy) The orbits are: f0g and R2 n f0g. The stabiliser of the point (1; 0) is the group N. Hence we have 2 R n f0g = G:(1; 0) ≈ G= StabG [(1; 0)] = G=N: Consequence As R2 n f0g and N are connected, G is connected. Remark (for the wine cellar) The linear action of G on R2 preserves the Lebesgue measure on R2 and induces a linear action on the full tensor algebra of R2. L. Flaminio (Université de Lille) SL2(R) Nantes, 2-7 juillet 2017 4 33 Consequence Since StabG [−I ] = K and since the sheet S is contractible from the fibration K ! G ! G=K ≈ S we get that K injects its π1 in G: π1(G) = π1(K) ≈ Z. Consequence Isomorphism PSL2(R) ≈ SO(1; 2)0.