Chapter33
          Hyperbolic plane
          In this chapter, we give background on the geometry of the hyperbolic plane.
          33.1  ⊲ The beginnings of hyperbolic geometry
                                                  1
          Wehave seen that the group of unit Hamiltonians H acts by rotations of Euclidean
          spaceandthereforebyisometriesoftheunitsphere,andthatinsphericalgeometrythe
          discrete subgroups are beautifully realized as classical finite groups: cyclic, dihedral,
          and the symmetry groups of the Platonic solids.
             Replacing H with M (R), the group SL (R) of determinant 1 matrices possesses a
                            2             2
          muchricher supply of discrete subgroups. In fact, PSL2(R) can be naturally identified
          with a circle bundle over the hyperbolic plane, and so the structure of quaternionic
          unit groups is naturally phrased in the language of hyperbolic geometry. Indeed, it
          wasworkonautomorphicfunctionsanddifferentialequationsinvariantunderdiscrete
          subgroups of PSL2(R) that provided additional early original motivation to study
          hyperbolic space: their study provides an incredibly rich interplay between number
          theory, algebra, geometry, and topology, with quaternionic applications. This interplay
          is the subject of the final parts of the text.
             In this chapter, we provide a rapid introduction to the hyperbolic plane. Hyperbolic
          geometry has its roots preceding the quaternions, in efforts during the early 1800s to
          understand Euclid’s axioms for geometry. Since the time of Euclid, there had been
          attempts to prove the quite puzzling parallel postulate (given a line and a point not
          on the line, there is a unique line through the point parallel to the given line) from
          the other four simpler, self-evident axioms for geometry. In hyperbolic geometry, the
          parallel postulate fails to hold—there are always infinitely many distinct lines through
          a point that do not intersect a given line—and so it is a non-Euclidean geometry.
             Theunderpinnings of what became hyperbolic geometry can be found in work by
          EulerandGaussintheirstudyofcurvedsurfaces(thedifferentialgeometryofsurfaces).
          It was then Lobachevsky and Bolyai who suggested that curved surfaces of constant
          negativecurvaturecouldbeusedinnon-Euclideangeometry,andfinallyRiemannwho
          generalized this to what are now called Riemannian manifolds. Klein coined the term
          “hyperbolic” for this geometry because its formulae can be obtained from spherical
          ©TheAuthor(s)2021                                             605
          J. Voight, Quaternion Algebras, Graduate Texts in Mathematics 288,
          https://doi.org/10.1007/978-3-030-56694-4_33
             606                                            CHAPTER33. HYPERBOLICPLANE
             geometry by replacing trigonometric functions by their hyperbolic counterparts. See
             [Sco83, §1] for a nice overview of the 2-dimensional geometries.
                 Hyperbolicgeometry,andinparticularthehyperbolicplane,remainsanimportant
             prototype for understanding negatively-curved spaces in general. Milnor [Milno82]
             gives a comprehensive early history of hyperbolic geometry; see also the survey by
             Cannon–Floyd–Kenyon–Parry[CFKP97],whichincludesanexpositionoffivemodels
             for hyperbolic geometry. (It is also possible to work out hyperbolic geometry in a
             mannerakintowhatEucliddidforhisgeometrywithoutaparticularmodel,following
             Lobachevsky [LP2010].)
                 For further references on hyperbolic plane geometry, see Jones–Singerman [JS87,
             Chapter 5], Anderson [And2005], Ford [For72], Katok [Kat92, Chapter 1], Iversen
             [Ive92, Chapter III], and Beardon [Bea95, Chapter 7]. There are a wealth of geometric
             results and formulas from Euclidean geometry that one can try to reformulate in the
             world of hyperbolic plane geometry, and the interested reader is encouraged to pursue
             these further.
             33.2    Geodesic spaces
             In geometry, we need notions of length, distance, and the straightness of a path. These
             notions are defined for a certain kind of metric space, as follows.
                                                                                 ∼
                 Let X be a metric space with distance ρ.Anisometry g : X −→ X is a bijective
             mapthatpreservesdistance,i.e., ρ(x, y) = ρ(g(x),g(y))forall x, y ∈ X.(Anydistance-
             preserving map is automatically injective and so becomes an isometry onto its image.)
             Theset of isometries Isom(X) forms a group under composition.
             33.2.1. A path from x to y, denoted υ: x → y, is a continuous map υ:[0,1] → X
             where υ(0) = x and υ(1) = y. (More generally, we can take the domain to be any
             compact real interval.) The length ℓ(υ) of a path υ is the supremum of sums of
             distances between successive points over all finite subdivisions of the path (the path
             is rectifiable if this supremum is finite). Conversely, if X is a set with a notion of
             (nonnegative) length of path, then one recovers a candidate (intrinsic)metricas
                                              ρ(x, y) =   inf ℓ(υ),                         (33.2.2)
                                                        υ:x→y
             a metric when this infimum exists (i.e., there exists a path of finite length x → y)for
             all x, y ∈ X. If the distance on X is of the form (33.2.2), we call X a length metric
             space or a path metric space, and by construction ℓ(gυ) = ℓ(υ) for all paths υ and
             g ∈ Isom(X).
             Example33.2.3. The space X = Rn with the ordinary Euclidean metric is a path
             metric space; it is sometimes denoted En as Euclidean space (to emphasize the role of
             the metric).
             33.2.4. If X is a path metric space and υ achieves the infimum in (33.2.2), then we say
             υ is a geodesic segment in X.Ageodesic is a continuous map (−∞,∞) → X such
             that the restriction to every compact interval defines a geodesic segment. If X is a path
            33.2. GEODESICSPACES                                                         607
            metric space such that every two points in X are joined by a geodesic segment, we say
            X is a geodesic space, and if this geodesic is unique we call X a uniquely geodesic
            space.
            33.2.5. If X is a geodesic space, then an isometry of X maps geodesic segments
            to geodesic segments, and hence geodesics to geodesics: i.e., if g ∈ Isom(X) and
            υ: x → y is a geodesic segment, then gυ: gx → gy is a geodesic segment. After all,
            g maps the set of paths x → y bijectively to the set of paths gx → gy,preserving
            distance.
            33.2.6. In the context of differential geometry (our primary concern), these notions
            can be made concrete with coordinates. Suppose U ⊆ Rn is an open subset. Then
            a convenient way to specify the length of a path in U is with a length element in
            real-valued coordinates. To illustrate, the ordinary metric on Rn is given by the length
            element                            
                                         ds :=   dx2 +···+dx2,
                                                   1          n
            so if υ:[0,1] → U is a piecewise continuously differentiable function written as
            υ(t) = (x (t),...,x (t)), then
                    1         n             
                                        ∫ 1    dx 2        dx 2
                                  ℓ(υ) =          1   +···+      n   dt              (33.2.7)
                                          0      dt            dt
            as usual.
                More generally, if λ: U → R>0 is a positive continuous function, then the length
            element λ(x)ds defines a metric (33.2.2)onU, as follows. The associated length
            (33.2.7) is symmetric, nonnegative, and satisfies the triangle inequality. To show that
            ρ(x, y) > 0 when x  y, by continuity λ is bounded below by some η>0onasuitably
            smallǫ ball neighborhoodof x notcontaining y,soeverypathυ: x → y hasℓ(υ) ≥ ǫη
            and ρ(x, y) > 0.
                In this context, we also have a notion of orientation, and we may restrict to isome-
            tries that preserve this orientation. We return to this in section 33.8, rephrasing this in
            terms of Riemannian geometry.
            Remark33.2.8. Themoregeneralstudyofgeometrybasedonthenotionoflengthina
            topologicalspace(theverybeginningsofwhicharepresentedhere)istheareaofmetric
            geometry.Metricgeometryhasseensignificantrecentapplicationsingrouptheoryand
            dynamical systems, as well as many other areas of mathematics. For further reading,
            see the texts by Burago–Burago–Ivanov [BBI2001] and Papadopoulous [Pap2014].
                In particular, geodesic spaces are quite common in mathematics, including com-
            plete Riemannian manifolds; Busemann devotes an entire book to the geometry of
            geodesics[Bus55].Uniquelygeodesicspacesarelesscommon;examplesincludesim-
            ply connected Riemannian manifolds without conjugate points, CAT(0) spaces, and
            Busemannconvexspaces.
                Thefollowing theorem nearly characterizes geodesic spaces.
            608                                          CHAPTER33. HYPERBOLICPLANE
            Theorem33.2.9 (Hopf–Rinow).LetX be a complete and locally compact length
            metric space. Then X is a geodesic space and every bounded closed set in X is
            compact.
            Proof. See e.g. Bridson–Haefliger [BH99, Proposition 3.7]).                       
            33.3    Upperhalf-plane
            We now present the first model of two-dimensional hyperbolic space (see Figure
            33.3.2).
            Definition 33.3.1. The upper half-plane is the set
                                  H2 := {z = x +iy ∈ C :Im(z) = y > 0}.
                                    Figure 33.3.2: Upper half-plane H2
                                                                   2
            Definition 33.3.3. The hyperbolic length element on H is defined by
                                               2        2
                                        ds :=   dx +dy = |dz| ;                        (33.3.4)
                                                    y        Imz
                Asdescribedin33.2.6, the hyperbolic length element induces a metric on H2, and
            this provides it with the structure of a path metric space.
                                        2
            Definition 33.3.5. The set H equipped with the hyperbolic metric is (a model for) the
            hyperbolic plane.
                                         2
            Remark 33.3.6. The space H can be intrinsically characterized as the unique two-
            dimensional (connected and) simply connected Riemannian manifold with constant
            sectional curvature −1.                                  2
                The hyperbolic metric and the Euclidean metric on H are equivalent, inducing
            the same topology (Exercise 33.1). However, lengths and geodesics are different under
            these two metrics, as we will soon see.
            33.3.7. The group GL2(R) acts on C via linear fractional transformations:
                                az+b                     
                                                     ab
                           gz =        ,    for g =         ∈ SL2(R) and z ∈ C;
                                cz+d                 cd