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# 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