2. The Geometry of Kleinian Groups
2.4 The γ Parameter Space
sufficient to determine whether various two-generator subgroups ofMare dis-
crete, [23]. Recent work has focused on developing methods for the classification of the cocompact arithmetic Kleinian groups, most specifically in the case of groups with two-elliptic-generators.
Theorem 2.3.9 (Theorem 2.6 in [23]). Let G = hf, gi be a Kleinian group,
withf a primitive elliptic element of ordern≥3,g an elliptic element of order
2, andγ6= 0, β. Then Gis a subgroup of an arithmetic Kleinian group if:
• Q(γ, β)has at most one complex place;
• γ is a root of a monic polynomialp(z)∈Z[β][z];
• if γ andγ are not real, then all other roots of p(z)are real and lie in the
interval (β,0);
• if γ is real, then all other roots of p(z) are real and lie in the interval
(β,0).
If Gis also of finite covolume, thenGis an arithmetic Kleinian group.
In the above theorem, the requirement on the field Q(γ, β) to have at most
one complex place can, for some cases, be described in terms of the polynomial
p(z). Specifically in the cases where n = 3,4,6 these criteria offer a much simpler description, see [23].
The interest in two-elliptic-generator arithmetic groups, noted in Theorem 2.3.9, is motivated by results proving that there are only finitely many arithmetic Kleinian groups generated by elliptic elements [39]. A full classification has already been determined for the non-cocompact case; and a description of 21 arithmetic generalised triangle groups, from the cocompact case, is given in [60]. The work here is related to a greater project on the classification of the two-elliptic-generator arithmetic groups.
2.4
The
γ
Parameter Space
Following Theorem 2.3.2 and the results of the previous section, it is common to focus on the investigation of two-generator Kleinian groups; and, more specifi- cally, the nature and “geometry” of their parameter sets when considered as a subspace ofC3.
The(p,2)-Commutator Plane
We consider only the two-generator Kleinian groupshf, giwheref is a primitive elliptic of orderp, andgis a primitive elliptic of orderq. In this case
par(hf, gi) = −4 sin π p ,−4 sin π q , γ ;
and one can consider the alternative parameter set
par(hf, gi) = (p, q, γ).
We follow the results of previous sections and consider the groupshf, giwhere
f is a primitive elliptic of orderp≥2 andg is an elliptic of order2. By fixing
p, attention can be restricted to the γ parameter and its parameter-space C; which is referred to as the (p,2)-commutator plane. While all non-zero values of the(p,2)-commutator plane may correspond to a two-generator group, only a subset of these will correspond to Kleinian groups.
Note that if(2,2, γ)corresponds to a discrete group, then it is either a cyclic or
dihedral group; therefore the cases whenp=q= 2 are commonly ignored.
In the casep >2, given a pointγ∈C∗one can then ask whether the parameter set (p,2, γ) corresponds to a Kleinian group? If it does, then does this group have finite covolume and is it arithmetic? And further, what can we say about the(p,2)-commutator plane in general?
Fig. 2.1: (3,2)-commutator plane with disc covering and fractal boundary.
(Figure 3.16 of of [62].)
Investigating these commutator planes is an active area of research, see [62], [23] and the references therein. The remainder of this section will give a brief
2.4. Theγ Parameter Space 23
overview of this research and explain the various results demonstrated on the
(3,2)-commutator plane shown in Figure 2.1; as seen in [62].
Comparable results and images have been shown for other (p,2)-commutator
planes, most specifically wherep= 3, ...,7. Similar images, and related results, can be found in [23] and [62].
We begin by highlighting two key features of all(p,2)-commutator planes.
2.4.1 Symmetry
The reflective symmetry, seen in Figure 2.1, follows from the results of Section 2.3.3 which imply a symmetry in discreteness results, on the(p,2)-commutator plane, about the linesIm(γ) = 0andRe(γ) =1
2β.
That is, if one of the following parameter sets corresponds to a (arithmetic) Kleinian group:
• (β,−4, γ), • (β,−4, β−γ), • (β,−4, γ), • (β,−4, β−γ);
then they all correspond to a (arithmetic) Kleinian group.
2.4.2 Bounds and Inequalities
By a fundamental domain construction argument based on the Klein combina- tion theorems, see [1], [24], it is known that for each commutator plane there exists a real numberγ0≥0 such that for all γ, if |γ|> γ0, then(p,2, γ)corre- sponds to a free group. This γ0-value is indicated by the circle seen in Figure 1.2.
Thus, when |γ| is large enough, every parameter set(p,2, γ)corresponds to a two-generator discrete grouphf, githat is isomorphic to the free product of the two cyclic subgroups hfi and hgi. In other words, (p,2, γ) corresponds to a group with presentation:
hf, g|fp, g2=Ii.
In addition to this free boundary |γ| > γ0, many other inequalities have been shown to exist between the various parameters. One of the earlier results in this area was Jørgensen’s inequality, which provides a necessary condition for non-elementary Kleinian groups.
Theorem 2.4.1 (Jørgensen’s Inequality, [33]). If hf, gi is a Kleinian group, then its parameters satisfy the following inequality
|β(f)|+|γ(f, g)|≥1.
This theorem has stimulated research in these inequalities, with many results having been determined with respect to the complex analysis of Möbius trans- formations; for example see [26], [27] and [57]. The inequality in Theorem 2.4.1 can easily be rearranged to have the form|γ| ≥rfor some positive real number
r; but, more specific to our setting, [28] provides the following inequality.
Theorem 2.4.2. Ifhf, giis a Kleinian group, andf org is an elliptic of order
n≥3 andγ6= 0, β, then |γ|≥2 cos 2π 7 .
These inequalities, and the free boundary, can be used to begin to describe the space of two-elliptic-generator Kleinian groups.
2.4.3 Disc Covering
Recent work, demonstrated in [23] and the references therein, has seen the devel- opment of methods which utilize a variety of inequalities, like those mentioned
above, and semi-group polynomials in the γ-parameter, to build a description
of theγ-plane. This description comprises of an iterative series of boundaries upon theγ-values corresponding to discrete groups within a(p,2)-commutator plane.
LetG=hf, gibe a discrete group where f andgare elliptic. Take
h=g◦fm1◦g−1◦fm2◦g◦...◦fmn◦g(−1)n,
where mi = ±1. Then hf, hi, as a subgroup of hf, gi, is discrete; by Theo-
rem 2.3.6, the group represented by(β(f),−4, γ(f, h))is discrete; and, further,
γ(f, h)is given by a semigroup polynomialt inγ(f, g)andβ(f).
Fixingp, the semigroup polynomial only has variable γ(f, g), which gives rise to an iterative process for the derivation of successive γ-values in the (p,2)- commutator plane.
γi+1=t(γi).
This iterative process is described in [28], and cannot have0as a limit. Further it cannot iterate into the regions determined by inequalities, such as those given
2.4. Theγ Parameter Space 25
in Section 2.4.2 above, to not hold discrete groups; with the exception of certain ‘exceptional’ values.
hf, gidiscrete ⇒ hf, hidiscrete hf, hinot discrete ⇒ hf, ginot discrete
IfRis one of the regions whereγ does not correspond to a discrete group, then neither does the set t−1(R) . This provides a disc covering method whereby the only valid Kleinian group values ofγ can be found inC\{∪t−1(R) :t∈τ}, whereτis the set of such semigroup polynomials; with the exception of certain, hard to determine, exceptional points.
The results of this disc covering method for the (3,2)-commutator plane, as
undertaken in [23], are shown in Figure 2.1; specifically the discs and the points contained within them.
Determining the regions R, and their exceptional points, can be difficult; but [23] notes that there exist polynomials in the arithmetic data for which valid, discreteγ-values are a root. Using this, one can determine arithmetic conditions which guarantee the discreteness of a group. The arithmetic theorems recently developed then reduce the search forγ-values down to a test on polynomials.
These methods have been used to eliminate large regions of possibleγ-values in a number of (p,2)-commutator planes. See [23] and its references, specifically [28], for more details and images of other disc coverings.
2.4.4 Fractal Boundary
Using SnapPea (see Section 3.1) for the experimental construction ofγ-words,
and eliminating the determinably free elements, progress has recently been made
in demonstrating a conjecturally fractal boundary beyond which all γ-values
correspond to free groups, [62].
This has been done through a consideration of (m,0) Dehn surgery upon hy-
perbolic 2-bridge links. These links correspond to hyperbolic orbifolds groups
generated by two order-m transformations. These groups are not free, so are
contained within the|γ0|circle, but are generally outside the disc covering given above. This work has given rise to a “computational boundary”, beyond which there are only free groups. In the (p,2)-commutator planes this is conjectured to be a fractal Jordan curve, and is observationally comparable to the Riley slice.
individual points surrounding the disc covering.
This work has greatly improved upon the previously mentioned |γ0| bounds,
and gives an indication towards the overall completeness of these disc covering methods.