Università degli Studi di Trieste ar Ts Archivio della ricerca – postprint Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions 1 1 Juan Pablo Díaz · Alberto Verjovsky · 2 Fabio Vlacci Abstract Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quater- nion division algebra H, we define several Kleinian discrete subgroups of PSL(2, H).We define first a Kleinian subgroup PSL(2, L) of PSL(2, H(Z)). This group is a general- ization of the modular group PSL(2, Z). Next we define a discrete subgroup PSL(2, H) of PSL(2, H) which is obtained by using Hurwitz integers. It contains as a subgroup PSL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and 1 1 describe the orb-ifold quotients HH/PSL(2, L) and HH/PSL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space. Keywords Modular groups · Arithmetic hyperbolic 4-manifolds · 4-Orbifolds · Quaternionic hyperbolic geometry Mathematics Subject Classification (2000) Primary 20H10 · 57S30 · 11F06; Secondary 30G35 · 30F45 Dedicated to Bill Goldman on occasion of his 60th birthday. B Alberto Verjovsky [email protected] Juan Pablo Díaz [email protected] Fabio Vlacci [email protected]fi.it 1 Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de Chamilpa, Código Postal 62210 Cuernavaca, Morelos, Mexico 2 Dipartimento di Matematica e Informatica DiMaI “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Florence, Italy 1 1 Introduction Since the time of Carl Friedrich Gauss and the foundational papers by Richard Dedekind and Felix Klein the classical modular group PSL(2, Z) and its action on the hyperbolic (complex) 1 upper half plane HC ={z ∈ C :(z)>0} have played a central role in different branches of mathematics and physics like number theory, Riemann surfaces, elliptic curves, hyperbolic geometry, theory of modular forms and automorphic forms, crystallography, string theory and others. Similarly, discrete subgroups of PSL(2, C) are very important in the construction of lattices to study of arithmetic hyperbolic 3-orbifolds (see, for instance, [20]) and many other fields of mathematics. We consider the action defined by isometries of the modular group PSL(2, Z) on the 1 hyperbolic plane HC. A fundamental domain is a triangle with one ideal point an two other vertices were the sides have an angle of π/3. This is the triangle with Coxeter notation (3, 3, ∞). See the Fig. 1. The modular group PSL(2, Z) is a subgroup of the group of 1 symmetries of the regular tessellation of HC whose tiles are congruent copies of the triangle (3, 3, ∞). We can describe the Cayley graph and a presentation of the group PSL(2, Z) in terms of 2 generators and 2 relations to obtain: PSL(2, Z) =a, b|a2 = (ab)3 = 1=Z/2Z ∗ Z/3Z. 2 1 2 The quotient O := HC/PSL(2, Z) has underlying topological space the plane R (or C)andO2 consists of two distinguished conical points. The local groups of the singular points are Z/2Z and Z/3Z modeled on groups of two and three elements which consist of hyperbolic rotations of angles π and 2π/3, respectively. The Euler characteristic of the orbifold O2 is −1/6. Thus a minimal surface Selberg cover is of order 6. Now we consider the action defined by isometries of the modular group PSL(2, Z[i]) 3 which is the Picard group related to the Gauss integers on the hyperbolic real 3-space HR. 3 3 3 The quotient O := HR/PSL(2, Z[i]) has underlying space the 3-space R . Its singular locus O3 is the 1-skeleton of a squared pyramid without the apex. The Euler characteristic of the orbifold O3 is 0. In this paper, we introduce two generalizations of the modular group in the four- dimensional setting of the quaternions and the rings of Lipschitz and Hurwitz integers (see [14,15],[19]) and then focus our attention to their actions on hyperbolic (quaternionic) half | |2 space H1 := {q ∈ H : (q)>0} with metric d q . We then explore new results which H ( q)2 give a very detailed description of the orbifolds related to the corresponding quaternionic modular groups. Secondly we define the group generated by translations by the imaginary 1 Fig. 1 A fundamental domain for the action of the modular group PSL(2, Z) on the hyperbolic plane HC 2 parts of Lipschitz integers, the inversion T and a finite group related to the Hurwitz units. We give a thorough description of its properties and the corresponding orbifolds. Furthermore, we also give a geometric description of the fundamental domains for the actions of the quaternionic modular groups and a detailed analysis of the topology and of the isotropy groups of the singularities of the orbifolds introduced. We study the corresponding modular Lipschitz and Hurwitz groups in the Lorentz–Minkowski model. 1 2 The quaternionic hyperbolic 4-space HH and its isometries 1 2.1 Isometries in the half-space model of the hyperbolic 4-space HH Consider the quaternions 2 2 2 H := {x0 + x1i + x2j + x3k : xn ∈ R, n = 0, 1, 2, 3, i = j = k =−1, ij =−ji = k}. If q = x0 + x1i + x2j + x3k ∈ H then (q) := x0 ∈ R, q := x0 − x1i − x2j − x3k ∈ H and |q|2 := qq ∈ R+. 1 Let HH := {q ∈ H : (q)>0} be the half-space model of the one-dimensional quaternionic hyperbolic space. This set is isometric to the hyperbolic real space in four 1 ∼ 4 4 dimensions, namely HH = HR ={(x0, x1, x2, x3) ∈ R : x0 > 0} with the element ( )2+( )2+( )2+( )2 ( )2 = dx0 dx1 dx2 dx3 of hyperbolic metric given by ds 2 where s measures length x0 along a parametrized curve. Even though the (natural) algebraic structures carried by the two sets are deeply different. Let GL(2, H) denote the general linear group of 2 × 2 invertible1 matrices with entries in the quaternions H. The next definitions can be found in [3,10,16]and[26]. ab Definition 2.1 For any A = ∈ GL(2, H), the associated real analytic function cd −1 FA : H ∪{∞}→H ∪{∞}defined by FA(q) = (aq + b) · (cq + d) is called the Möbius transformation associated with A. −1 −1 We set FA(∞) =∞if c = 0, FA(∞) = ac if c = 0andFA(−c d) =∞. Let F := {FA : A ∈ GL(2, H)} the group of Möbius transformations. Definition 2.2 Let SL(2, H) be the special linear group which consists of matrices in GL(2, H) with Dieudonnè determinant 1. Then : GL(2, H) → F defined as (A) = FA is a surjective group antihomomorphism with ker() ={tI : t ∈ R \{0}}. Furthermore, the restriction of to the special linear group SL(2, H) is still surjective and whose kernel is {±I}. 1 Let M 1 be the set of Möbius transformations that leave invariant H . Any transformation HH H 1 F ∈ M 1 is conformal and preserves orientation, moreover is an isometry of H .We A HH H 1 1 conclude that M 1 is isomorphic to the groups Conf+(H ) and Isom+(H ) of conformal HH H H 1 diffeomorphisms and isometries orientation–preserving of the half-space model HH (see [2] and [1]). Moreover, the group M 1 acts by orientation-preserving conformal transformations HH 1 on the sphere at infinity of the hyperbolic 4-space defined as ∂HH := {q ∈ H : (q) = ∼ 3 ∼ 1 ∼ 1 0}∪{∞}. In other words M 1 = Conf+(S ) = Conf+(H ) = Isom+(H ). HH H H 1 By this we mean that a 2 × 2 quaternionic matrix A has a right and left inverse; in [4] it is shown that this is equivalent for A to have non zero Dieudonné determinant (see [3]). 3 Finally we recall the conditions found by Ahlfors (see [2])andthenappliedbyBisiand Gentili in the next form (see [4]): Proposition 2.3 [Ahlfors conditions] The subgroup M 1 of P SL(2, H) can be charac- HH terized as the group induced by matrices which satisfy one of the following (equivalent) conditions: ⎧ ⎪ ⎪ = ab , , , ∈ H : t = = 01 , ⎪ A a b c d A KA K with K ⎪ cd 10 ⎪ ⎨⎪ ab A = a, b, c, d ∈ H : (ac) = 0, (bd) = 0, bc + da = 1 , ⎪ cd ⎪ ⎪ ⎪ ⎪ ab ⎩ A = a, b, c, d ∈ H : (cd) = 0, (ab) = 0, ad + bc = 1 . cd 1 2.2 The affine subgroup A(H) of the isometries of HH Consider now the affine subgroup A(H) of M 1 consisting of transformations which are HH λab induced by matrices of the form with |a|=1,λ>0and (ba) = 0. The group 0 λ−1a A(H) is the maximal subgroup of M 1 which fixes the point at infinity and its a Lie group HH of real dimension 7. Each matrix in A(H) acts as a conformal transformation on the hyperplane at infinity 1 ∂HH. Moreover A(H) is the group of conformal and orientation preserving transformations acting on the space of pure imaginary quaternions at infinity which can be identified with R3 so that A(H) is isomorphic to the conformal group Conf+(R3). 1 2.3 Isotropy subgroup of the isometries of HH which fixed one point αβ Let K := ∈ M 1 be the subgroup of symmetric matrices in M 1 .Forthematrix βα HH HH αβ |α|2 +|β|2 = (αβ) = βα the conditions 1and 0 are equivalent to Ahlfors conditions in 1 Proposition 2.3.