Modular groups over the quaternions Alberto Verjovsky* North-American Workshop in Holomorphic Dynamics May 27-June 4, 2016 Cancún, México Celebrating John Milnor’s 85th birthday * Instituto de Matemáicas UNAM June 11, 2016 Alberto Verjovsky Modular groups HOLOMORPHIC DYNAMICS IN PARADISE Alberto Verjovsky Modular groups Figure 14. A fundamental domain for the action of the modular group PSL(2; ) on the hyperbolic plane H2 and the Z R corresponding tessellation . Since the time of Carl Friedrich Gauss one of the most fascinating and important objects in mathematics is the modular group and its action on the upper half-plane of complex numbers. a b SL(2; ) := : a; b; c; d 2 ; ad − bc = 1 Z c d Z az + b z 7! cz + d PSL(2; Z) = SL(2; Z)=fI; −Ig . Alberto Verjovsky Modular groups Complex Modular Group The modular group can be shown to be generated by the two transformations T : z 7! −1=z S : z 7! z + 1 so that every element in the complex modular group can be represented (in a non-unique way) by the composition of powers of T and of S. Geometrically, T represents inversion in the unit circle followed by reflection with respect to the origin, while S represents a unit translation to the right. Alberto Verjovsky Modular groups The generators T and S obey the relations T 2 = I and (TS)3 = I and it can be shown that these are a complete set of relations, so the modular group has the following presentation: fT ; S j T 2 = I; (TS)3 = Ig ∼ PSL(2; Z) = Z2 ? Z3 Alberto Verjovsky Modular groups Figure 14. A fundamental domain for the action of the modular group PSL(2; ) on the hyperbolic plane H2 and the Z R corresponding tessellation . Alberto Verjovsky Modular groups Alberto Verjovsky Modular groups Let H = fx0 + x1i + x2j + x3k : xn 2 R; n = 0; 1; 2; 3g i2 = j2 = k2 = −1; ij = −ji = k: be the division algebra of the quaternions. Then H is can be identified in the natural way with R4. We define the hyperbolic half-space H1 := fq 2 : <(q) > 0g. H H Alberto Verjovsky Modular groups Definition Let H1 be the half-space model of the one-dimensional H quaternionic hyperbolic space 1 HH := fq 2 H : <(q) > 0g: It can be identified with the hyperbolic space of dimension four H4 : R 1 ∼ 4 4 HH = HR = f(x0; x1; x2; x3) 2 R : x0 > 0g with the element of hyperbolic metric given by 2 2 2 2 2 (dx0) +(dx1) +(dx2) +(dx3) (ds) = 2 where s measures length x0 along a parametrized curve Alberto Verjovsky Modular groups Definition a b For any A = 2 GL(2; ), the associated real analytic c d H function FA : H [ f1g ! H [ f1g defined by −1 FA(q) = (aq + b) · (cq + d) (1) is called the linear fractional transformation associated to A. We −1 set FA(1) = 1 if c = 0, FA(1) = ac if c 6= 0 and −1 FA(−c d) = 1. Alberto Verjovsky Modular groups Let F := fFA : A 2 GL(2; H)g the set of linear fractional transformations. 2 Since H × H = H := (q0; q1): q0; q1 2 H as a real vector space is R8, the group GL(2; H) can be thought as a subgroup of GL(8; R). Using this identification we define: Definition Let SL(2; H) := SL(8; R) \ GL(2; H) be the special linear group and PSL(2; H) := SL(2; H)={±Ig; where I denotes the identity matrix, the projective special linear group. Alberto Verjovsky Modular groups Theorem The set F is a group with respect to the composition operation and the map Φ: GL(2; H) ! F defined as Φ(A) = FA is a surjective group antihomomorphism with ker(Φ) = ftI : t 2 R n f0gg. Furthermore, the restriction of Φ to the special linear group SL(2; H) is still surjective and has kernel {±Ig. Alberto Verjovsky Modular groups Let B denote the open unit ball in H and MB the set of linear transformations that leave invariant B or Möbius transformations. This is the set MB := fF 2 F : F(B) = Bg: 1 0 H = and 0 −1 t Sp(1; 1) := fA 2 GL(2; H): AHA¯ = Hg: Alberto Verjovsky Modular groups There is an interesting characterization of these Möbius transformations: Theorem Given A 2 GL(2; H), then the linear fractional transformation FA 2 MB if and only if there exist u; v 2 @B, q0 2 B (i.e., juj = jvj = 1 and jq0j < 1) such that −1 FA(q) = v(q − q0)(1 − q0q) u (2) for q 2 B. In particular, the antihomomorphism Φ can be restricted to a surjective group antihomomorphism Φ: Sp(1; 1) !MB whose kernel is {±Ig. Alberto Verjovsky Modular groups Proposition The Poincaré distance dB given by: 4jdqj2 (1 − jqj2)2 in B is invariant under the action of the group MB of Möbius transformations. In other words: M = Isom+ (B): B dB Alberto Verjovsky Modular groups The compactification Hb := H [ f1g of H can be identified with 4 S via the stereographic projection. The elements of MB act conformally on the 4-sphere with respect to the standard metric and they also preserve orientation and preserve the unit ball. Therefore we conclude that 4 MB ⊂ Conf+(S ); 4 where Conf+(S ) is the group of conformal and orientation-preserving diffeomorphisms of the 4-sphere S4. As 4 a differentiable manifold, Conf+(S ) is diffeomorphic to SO(5) × H5 with H5 = f(x ; x ; x ; x ; x ) 2 5 : x > 0g, so R R 1 2 3 4 5 R 1 4 Conf+(S ) has real dimension 15. Alberto Verjovsky Modular groups Let us give a different description of this group. We recall that S4 can be thought of as being the projective quaternionic line P1 =∼ 4. This is the space of right quaternionic lines in 2, i.e., H S H subspaces of the form 2 Lq := f(q1λ, q2λ, ): λ 2 Hg ; (q1; q2) 2 H n f(0; 0)g: We recall that H2 is a right module over H and the action of GL(2; H) on H2 commutes with multiplication on the right, i.e. for every λ 2 H and A 2 GL(2; H) one has, ARλ = RλA where Rλ is the multiplication on the right by λ 2 H. Thus GL(2; H) carries right quaternionic lines into right quaternionic lines, and in this way an action of GL(2; ) on P1 =∼ 4 is H H S defined. Alberto Verjovsky Modular groups Any F 2 lifts canonically to an automorphism F of P3 , the A F fA C complex projective 3-space and the map Ψ: FA 7! FfA injects F into the complex projective group PSL(4; C) := SL(4; C)={±Ig. 4 MB ⊂ PSL(2; H) := SL(2; H)=ftI; t 6= 0g ' Conf+(S ): Alberto Verjovsky Modular groups Poincaré extension to the fifth dimension. As we have seen before the quaternionic projective line P1 can H be identified with the unit sphere S4 in R5 and therefore S4 is the boundary of the closed unit ball D5 ⊂ R5. As usual, we identify the interior of D5 with the real hyperbolic 5-space H5 . R Since PSL(2; ) acts conformally on 4 =∼ P1 , by Poincaré H S H Extension Theorem each element γ 2 PSL(2; H) extends canonically to a conformal diffeomorphism of D5 which restricted to H5 is an orientation preserving isometry γ~ of the R open 5-disk B5 with the Poincaré’s metric. Alberto Verjovsky Modular groups Reciprocally, any orientation preserving isometry of H5 extends R canonically to the ideal boundary R4 [ f1g as an element of PSL(2; H). Thus the map γ 7! γ~ is an isomorphism and PSL(2; ) = Isom H5 . H + R This is the connection between twistor spaces, conformal geometry, hyperbolic geometry in 5-dimensions and complex Kleinian groups. Alberto Verjovsky Modular groups Isometries in the half-space H1 . H Consider H1 be the half-space model of the one-dimensional H quaternionic hyperbolic space 1 HH := fq 2 H : <(q) > 0g: Via the Cayley transformation Ψ: B ! H1 defined as H Ψ(q) = (1 + q)(1 − q)−1 one can show explicitely that the unit ball B of is diffeomorphic to H1 and introduce a Poincaré H H distance in H1 in such a way that the Cayley transformation H Ψ: B ! H1 is an isometry; moreover the Poincaré distance in H H1 is invariant under the action of the group Ψ Ψ−1 which is H MB denoted by MH1 . H Alberto Verjovsky Modular groups Since the unit ball B in H can be identified with the lower 4 hemisphere of S and any transformation FA 2 MB is conformal and preserves orientation, we conclude that (see also [?, ?]) 1 MB ' Conf+(HH) where Conf (H1 ) represents the group of conformal + H diffeomorphisms orientation–preserving of the half-space model H1 . H Definition Let MH1 the subgroup of PSL(2; H) whose elements are H associated to invertible linear fractional transformations which preserve H1 . H Alberto Verjovsky Modular groups According to Lars V. Ahlfors it was Karl Theodor Vahlen who iintroduces in 1901 the idea of using the Clifford numbers to define Möbius groups of 2 by 2 matrices with entries in the Clifford numbers (Clifford matrices) acting on hyperbolic spaces. In 1984 Ahlfors pushes forward the idea of Vahlen and considers groups of 2 by 2 Clifford matrices and gives the necessary and sufficient conditions to leave invariant a half-space and the corresponding hyperbolic metric.