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 rather than with the whole space 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 Kleinian groups are groups which have a finite orbit in .. 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 (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 and (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.)
The limit set of a Fuchsian / Kleinian group is the set of limit points of -orbits of .
Note, since properly discontinuously, we have .
Proposition: is the smallest -invariant subset of .
Proof: It is clear that is -invariant; conversely, -invariant subset of must contain all of the limit points of the orbits of , i.e. must contain all of .
If , then is elementary.
Proposition: If then is either all of , 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: 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 is . It is the largest open subset of the boundary on which acts properly discontinuously.
Proposition: For a finitely-generated Kleinian group, if is non-empty, then has one, two, or infinitely many components, each of which is either simply or infinitely connected. There are additional rigidity results on if preserves one or more of the components.
(Marden, Lemma 2.4.2)
Remark: in the Fuchsian case with , each component of 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 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 determined by the intersection of the hyperbolic half-spaces .
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 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 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 .
(The rank of a cusp is the smallest number of generators of the parabolic subgroup of 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 or , resp.): given a Fuchsian / Kleinian group , the Ford fundamental region is the closure of the set of points in exterior to all isometric circles / hemispheres of elements of .
For Fuchsian, (Katok, Theorem 3.3.5.)
In the more general Kleinian case, is a limit of Dirichlet fundamental polyhedra as . (Marden, Lemma 3.5.2.)
- Svetlana Katok, Fuchsian Groups
- Albert Marden, Outer Circles