The many faces of the hyperbolic plane

\mathbb{H}^2 is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

  1. It is diffeomorphic to \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) as a topological space (or, indeed, isometric as a Riemannian manifold, if we give \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) a left-invariant metric): to show this, we note that \mathbb{H}^2 has isometry group \mathrm{SL}(2,\mathbb{R}) / \pm\mathrm{id}, and the subgroup of isometries which stabilize any given p \in \mathbb{H}^2 is isomorphic to \mathrm{SO}(2).
  2. Since any positive-definite binary quadratic form is given by a symmetric 2-by-2 matrix with positive eigenvalues, and since the group of linear transformations on \mathbb{R}^2 preserving any such given form is isomorphic to \mathrm{O}(2), \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the space of positive-definite binary quadratic forms of determinant 1, via the map from \mathrm{SL}(2,\mathbb{R}) to the symmetric positive-definite 2-by-2 matrices given by g \mapsto g^T g.
  3. \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2 is also the moduli space of marked Riemann surfaces of genus 1, i.e. the Teichmüller space Teich(S) of the torus S. One way to prove this is to note that any such marked surface is the quotient of \mathbb{R}^2 by a \mathbb{Z}^2 action; after a suitable conformal transformation, we may assume that the generators of this \mathbb{Z}^2 act as z \mapsto z + 1 and z \mapsto z + \tau for some z \in \mathbb{H}^2 (in the upper half-plane model.) But now \tau is the unique invariant specifying this point in our moduli space.
  4. Since any marked Riemann surface of genus 1 has a unique flat metric (inherited as a quotient manifold of the Euclidean plane), \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the moduli space of marked flat 2-tori of unit area.
  5. Since there is a unique unit-covolume marked lattice associated to each marked complex torus in the above, \mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) is also the space of marked lattices in \mathbb{R}^2 with unit covolume.
  6. Note we may go directly between marked Riemann surfaces and quadratic forms by considering intersection forms.

Fuchsian vs. Kleinian

Or, all the fun you could have in one additional dimension.

The former, of course, may be viewed as a special case of the latter; the more general Kleinian case shares many, but not all, of the characteristics of the simpler Fuchsian case. In particular, there is still a broad three-way classification of elements as elliptic, parabolic, or hyperbolic / loxodromic, and many arguments still proceed by examining these elements and their fixed loci.

The additional dimension introduces some combinatorial / geometric complications into these arguments; in particular, it is sometimes easier and more enlightening in the three-dimensional case to work purely at the boundary \partial\mathbb{H}^3 \cong S^2 rather than with the whole space \mathbb{H}^3 when making fixed-point (or other) arguments.

Here we collect a few facts about Fuchsian / Kleinian groups, mostly in order to highlight some things that stay the same, and other things that are new, when we go up from 2 to 3 dimensions in this case.

In the below, n is 2 in the Fuchsian case, and 3 in the Kleinian case.

Elementary (sub)groups

Elementary Kleinian groups are groups which have a finite orbit in \partial\mathbb{H}^n.. Any elementary group has a finite orbit in the boundary of size at most 2 (see result on the limit set below.)

Every subgroup of \mathrm{PSL}(2,\mathbb{R}) (and hence, a fortiori, any Fuchsian group) consisting of only elliptic elements (besides the identity) has a common fixed point, and hence is elementary (indeed, cyclic.)

More generally, every elementary Fuchsian group is cyclic, or conjugate to a group generated by g(z) = kz and h(z) =- -1/z (i.e. a hyperbolic and a elliptic switching the two ends of the axes.)

(Proof: Katok, Theorems 2.4.1 and 2.4.3)

Kleinian elementary groups include

  • the dihedral, tetrahedral, octahedral, and icosahedral groups;
  • the finite extension of a rank one or two free abelian group of parabolics by elliptics; there are finitely many possiblities for the orders of the extension, by ___;
  • a cyclic loxodromic group, possibly extended by the elliptic with the same axis, and possibly extended again by an order-two elliptic exchanging the endpoints of the axis.

(Proof: Marden, section 2.3; attributed to Ford 1929.)

Limit sets

The limit set \Lambda(\Gamma) of a Fuchsian / Kleinian group \Gamma is the set of limit points of \Gamma-orbits of \mathbb{H}^n.

Note, since \Gamma \curvearrowright \mathbb{H}^n properly discontinuously, we have \Lambda(\Gamma) \subset \partial\mathbb{H}^n.

Proposition: \Lambda(\Gamma) is the smallest \Gamma-invariant subset of \partial\mathbb{H}^n.

Proof: It is clear that \Lambda(\Gamma) is \Gamma-invariant; conversely, \Gamma-invariant subset of \partial\mathbb{H}^n must contain all of the limit points of the orbits of \Gamma, i.e. must contain all of \Lambda(\Gamma).

If |\Lambda(\Gamma)| \leq 2, then \Gamma is elementary.

Proposition: If |\Lambda(\Gamma)| > 2 then \Gamma is either all of \partial\mathbb{H}^n, or is a perfect (hence uncountable) nowhere dense subset of the boundary

(Proof: Katok, Theorem 3.4.6 and Marden, Lemma 2.4.1)

Proposition: \Lambda(\Gamma) is the closure of the set of hyperbolic / loxodromic fixed points

(Proof: Katok, Theorem 3.4.4; Marden, Lemma 2.4.1)

Domains of discontinuity

The domain of discontinuity \Omega(\Gamma) is \partial\mathbb{H}^n \setminus \Lambda(\Gamma). It is the largest open subset of the boundary on which \Gamma acts properly discontinuously.

Proposition: For a finitely-generated Kleinian group, if \Omega(\Gamma) is non-empty, then \Omega(\Gamma) has one, two, or infinitely many components, each of which is either simply or infinitely connected. There are additional rigidity results on \Omega(\Gamma) if \Gamma preserves one or more of the components.
(Marden, Lemma 2.4.2)

Remark: in the Fuchsian case with n=2, each component of \Omega(\Gamma) is (trivially) an open interval, and hence simply-connected, and the result on the number of components is a straightforward corollary of the Proposition above characterizing the structure of \Lambda(\Gamma) when it has cardinality larger than 2.

Proof: Uses Ahlfors finiteness in the general case (see e.g. Marden, Lemma 2.4.2 and Chapter 3.)

Contrasted with the relatively elementary arguments above, this is a rather deeper argument.

Fundamental regions and geometric finiteness

A Fuchsian / Kleinian group has Dirichlet fundamental regions / polyhedra D_p(\Gamma) determined by the intersection of the hyperbolic half-spaces H_p(T) =\{z \in \mathbb{H}^n : d(z,p) \leq d(z,T(p)) \}.

This is connected, convex, and locally-finite polygon / polyhedron—but not necessarily finite-sided. For Fuchsian groups, we have the following

Proposition: A Fuchsian group \Gamma is geometrically finite iff it is finitely-generated

(Proof: Forward direction–Katok, Theorem 3.5.4; reverse direction–Katok, Theorem 4.6.1)

Siegel’s Theorem: Any Fuchsian group of finite covolume is geometrically finite

The analogous results for general Kleinian groups are more subtle:

Theorem (Marden): A Kleinian group \Gamma is geometrically finite iff it is cocompact, except possibly for a finite number of rank one and rank two cusps, where the rank one cusps correspond to pairs of punctures on the boundary of \mathbb{H}^3 / \Gamma.

(The rank of a cusp is the smallest number of generators of the parabolic subgroup of \Gamma fixing the corresponding parabolic fixed point.)

Lemma (Wielenberg) Any Kleinian group of finite covolume is geometrically finite.

Proof: uses the thick-thin decomposition, and Marden’s theorem above.

There is also the Ford fundamental region / polyhedron, defined in terms of isometric circles / hemispheres (in \mathbb{H}^2 or \mathbb{H}^3, resp.): given a Fuchsian / Kleinian group \Gamma, the Ford fundamental region F(\Gamma) is the closure of the set of points in \mathbb{H}^n exterior to all isometric circles / hemispheres of elements of \Gamma.

For \Gamma Fuchsian, F(\Gamma) = D_0(\Gamma) (Katok, Theorem 3.3.5.)

In the more general Kleinian case, F(\Gamma) is a limit of Dirichlet fundamental polyhedra D_p(\Gamma) as p \to \partial\mathbb{H}^3. (Marden, Lemma 3.5.2.)

Offline references

  • Svetlana Katok, Fuchsian Groups
  • Albert Marden, Outer Circles