Abstract
A two-generator Kleinian group can be naturally associated with a discrete group with the generator of order two and where This is useful in studying the geometry of the Kleinian groups since will be discrete only if is, and the moduli space of groups is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of . The dimension reduction here is realised by a projection of principal characters of the two-generator Kleinian groups. In applications, it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen’s results on algebraic convergence.
1. Introduction
It is shown in [1] that every two-generator Kleinian group is determined uniquely up to conjugacy by its principal character, i.e., the triple of complex parameters
Here, where is the multiplicative commutator. Thus, the space of the two-generator Kleinian groups can be identified as a subspace of the three complex dimensional space . We define via the mapping
Things are set up so the group where is the identity of the group.
Applying [1, Lemma 2.29], given a two-generator Kleinian group with principal character with , there is a two-generator discrete group with principal character , i.e., is elliptic of order two. However, we can waive the condition (see Lemma 8 below).
If the group is Kleinian (i.e., not virtually abelian), then there are a series of inequalities on the entries of the principal character which need to hold. One such is Jørgensen’ inequality, Conversely, it is often easier to establish new discreteness conditions of two-generator groups with one generator of order two since we know of many polynomial trace identities that yield a polynomial semigroup action on the associated moduli space, and the dynamics of this action can be used to investigate discreteness of groups. Let us give one such example leading to Jørgensen’s inequality to motivate our subsequent results. We note which we write as Following [2], we write and Repeated application of this identity followed by projection yields the sequence giving a sequence of traces of commutators in the original discrete group where the series of commutators is
It is usually not too difficult to prove such sequences cannot be infinite and contain convergent subsequences. Jørgensen’s inequality is implied by the assertion that ; hence, . Every such trace identity yields a new inequality and the semigroup structure is clearly seen in general by where for example, from the trace identity
In another direction, the ergodicity of the polynomial semigroup acting in the complement of the Riley slice, is used to prove the supergroup density and the existence of the unbounded Nielsen classes of generating pairs [3]. Note that this is not the usual definition of the Riley slice but is known to be equivalent.
In order to further these studies, we establish a number of useful results in this article. Dynamically, the elementary subgroups are sinks for the polynomial semigroup action on a slice and are isolated within the subset corresponding to discrete groups (apart from trivial abelian and dihedral examples). Thus, after we observe elementary properties of a two-generator discrete group with principal character in §3, we list the possible principal characters in the first table in §4 representing. Then, we prove the two complex dimensional slice space is closed in §5. Then, is the projection of the three complex dimensional moduli space to . Applications of this set being closed are in the generalizations of Jørgensen’s inequality quantifying the isolated nature of the elementary groups in the moduli space of discrete groups [4].
2. Preliminaries
Let be the Möbius group of the normalized orientation preserving Möbius transformations on the Riemann sphere and let be the projective special linear topological quotient group: where Then, these two groups are topologically isomorphic, and so there are at least two different ways of thinking about groups throughout this paper, as either subgroups of or subgroups of Each of these groups has its own topology; however, the topological isomorphism shows us that the concept of discreteness is the same.
A subgroup of is called a Kleinian group if it is a nonelementary discrete group (see [5, 6]). Here, a group is elementary if it is virtually abelian, i.e., a finite extension of an abelian group. The elementary groups are classified [5, Section 5.1]. A subgroup of is nonelementary if the limit set, contains infinitely many points. Notice from [7] that a group is Kleinian if and only if every two-generator subgroup of is discrete. It is this characterization of the Kleinian groups that motivates a focus on two-generator groups.
The principal character of a representation of the two-generator group conveniently encodes various other geometric quantities. One of the most important is the complex hyperbolic distance between two axes and of the two elliptic or loxodromic Möbius transformations and where is the hyperbolic line in the hyperbolic -space joining the fixed points of and in , δ is the hyperbolic distance between those two axes, and is the holonomy of the transformation whose axis contains the common perpendicular between those two axes and moves to Thus,
The parameter describes the conjugacy class of the Möbius transformation in which lies.
Lemma 1. If an element of is not the identity, then (a) is parabolic if and only if .(b) is elliptic if and only if .(c) is loxodromic if and only if .
In addition, the parameter gives the order of each elliptic element.
Lemma 2. If an element of is not the identity, then is elliptic of order if and only if
We use the following easy lemma: a group is Kleinian if and only if it is virtually Kleinian, i.e., has a Kleinian subgroup of finite index.
Lemma 3. Let be a finite index subgroup of . Then, is Kleinian if and only if is Kleinian.
The proof follows directly from the fact that if has finite index in then it has a coset decomposition where for some Thus, any sequence contains a subsequence lying in one coset.
3. Slicing Spaces of Two-Generator Discrete Groups
Suppose that is two-generator discrete subgroup of with and . Then, Gehring and Martin [1] state that there exist elliptics and of order two such that and are discrete with the related two-generator groups principal characters and , respectively.
Notice that implies and have a common fixed point on and hence is elementary (see [5], Theorem 4.3.5 and [1], Identity (1.5)). Thus, the condition is necessary for Kleinian groups. We now consider the other restriction .
We start with our observation of the properties of a two-generator discrete group with the principal character which has one generator of order two.
Lemma 4. Let be a discrete group with principal character If then is an elementary discrete group, which is isomorphic to a dihedral group for some .
Proof. There are two cases to consider. If is parabolic, then by Lemma 1, and hence, and have a common fixed point which we may assume by conjugacy is . Then, is a subgroup of the upper-triangular matrices, where and the result follows. We may now assume and have the form Computing the commutator, and hence, and . The assumption now implies that . Since . Now, so which gives . Then, and Thus, interchanges the fixed points and of So has a finite orbit , and hence, it is an elementary discrete group, [5]. Computation shows the conjugate Thus, is a dihedral group , for some from Theorem 9.
We recall that the Riemann sphere is the boundary of the hyperbolic -space
Lemma 5. Let be a discrete group with the principal character then is isomorphic to one of the following elementary discrete groups: (a)The dihedral group if or in .(b)A dihedral group for some if in .(c)A parabolic group, if .
Proof. (1) Let be the common perpendicular between two axes and Since and are elliptic of order two, they are the rotations of order two about their axes and in respectively. And each of and interchanges the ending points of and fixes setwise; hence, is elementary. Furthermore, the product fixes the ending points of and fixes setwise, so the axis It follows that is loxodromic Next, from the Fricke identity (see [8]), we have Since and the previous identity (19) gives and by Lemma 4, is a dihedral group. Notice that is loxodromic with infinite order; hence, (2) in Since and have a common fixed point (say ), is elementary. Since is discrete, the axes of elliptics both of order two must intersect at an angle for some and the product is a rotation about with the oriented angle for any and some where is perpendicular to and and passing through Therefore, is elliptic of order Notice that and are rotations of order two and their rotation axes and meet in we have It follows that (3) in We may assume by conjugacy that is the common fixed point of and in . Then, and fix and hence, and are vertical hyperbolic lines. Notice that both and are rotations with the angles at distinct centers. Since the sum of two rotation angles is the product is a translation, and hence, is parabolic fixing Thus, the classification of elementary groups [5, Section 5.1], is a group of the Euclidean similarities of We next note the following elementary result.
Lemma 6. Let be a two-generator group with principal character then is an index two subgroup of .
Then, we have the following.
Lemma 7. Suppose that is a two-generator Kleinian group with principal character If then
Proof. Since is a two-generator Kleinian group, , and hence, and cannot share any fixed points, so Since , there are two cases for us to consider.
If has one element, say , then But Thus,
If has two elements, say , then and
If then by [5, Theorem 5.1.2] and cannot share exactly one fixed point, and hence, Thus, the only possibility is that interchange the fixed points of , so and which implies that fixes and but they are not fixed by We conclude that has at least three fixed points and this implies that is the identity, has order two and
In [1, Remark 2.6] it is shown that if is a two-generator discrete group, then there are two elliptic elements and of order two such that
In particular, if is elliptic or loxodromic, then the axes and meet at right angles to one and other, and they bisect the common perpendicular between the axes and
Lemma 8. Let be a Kleinian group with principal character Then, there are two elliptic elements and of order two such that and are discrete -extension of with principal characters and , respectively.
Proof. Note is a subgroup of both and Suppose that the principal characters for and are and respectively. Using [1, Lemma 2.1] We have two quadratic equations Solving these two quadratic equations, or and or As [9, Identity (6.9)] shows both possibilities occur, so Thus, after relabeling, the principal characters for and are and respectively. This completes the proof.
4. Exceptional Set of Principal Characters
Now, the natural question is to determine whether two discrete groups and in Lemma 8 are actually Kleinian. Since both and are -extension of by Lemma 3, we only need to decide if is a Kleinian subgroup. First, we list the possible principal characters in Tables 1–3 representing two-generator elementary discrete groups. The set of principal characters occurring in these three tables is called the exceptional set of principal characters.
Theorem 9. Let be an elementary discrete group of , then is isomorphic to one of the following groups, where
A cyclic group A dihedral group The group or A Euclidean translation group
A Euclidean triangle group or or A finite spherical triangle group , or or
Let and be elliptic of order in and let be the angle subtended at the origin between and , and hence, the hyperbolic distance between those two axes is Then, can be computed by using [9, Lemmas 6.19, 6.20, and 6.21], the parameters and , and the commutator parameter is , see [1]. With these elementary observations, some spherical trigonometry, and Theorem 9 about the classification of the elementary discrete groups, we obtain the following Tables 1 and 2, which list the possible principal characters (including the cases in the thesis [10]) of two-generator elementary discrete groups with nonzero commutator parameters They are either the dihedral groups (if and or the finite spherical triangle groups , and
Notice that the angle between intersecting axes of elliptics of order in a discrete group is always either when they meet on the Riemann sphere or This yields the additional parameter for the elementary group when generated by two elements of order . Furthermore, the angle between intersecting axes of elliptics of order in a discrete group is either or its complement . After possibly taking powers of the generator of order , the three additional principal characters can be obtained in the following Table 3.
The classification of the elementary discrete groups (Theorem 9) plays an important role in listing the tables above. We make the following additional remarks. (i)The axes of elliptics both of order two can intersect at an angle for any and giving the dihedral group with principal character (see Lemma 5).(ii)The axes of elliptics elements of orders and , , can meet at right angles. In this case, the principal character of dihedral group is (iii)The axes of elliptics of order and () in a discrete group meet on , i.e., meeting with angle , if and only if
For all of these groups In particular, the Euclidean triangle groups , , and cyclic groups have Also, the Euclidean translation groups have
Lemma 10. Let be a Kleinian group with the principal character that is not one of those exceptional groups listed in Table 1. Then, the subgroup is Kleinian.
Proof. We need to show that the discrete subgroup of the Kleinian group can not be elementary. Let and By Lemma 8 there is an elliptic of order two such that is a discrete group containing with index two and the principal character for is . By the hypothesis, is not one of those exceptional groups listed in Table 1; then, is not a finite spherical triangle group and Thus, can only be elementary if i.e., or Since is Kleinian, So it can only be , and hence, it is the dihedral group. In this case, since is not of order two, . In the case it gives , and hence, is elliptic of order two (Lemma 7) fixing or interchanging the fixed points of in . In the case might be a power of . In either case is not Kleinian, a contradiction.
The following theorem guarantees that is a Kleinian group if is not elliptic of order
Theorem 11. Suppose that is a Kleinian group. If is not elliptic of order , then the subgroup is Kleinian.
Proof. Suppose that is elliptic of order two. Then, is a subgroup of index two in and Lemma 3 ensures that is also Kleinian. Suppose that the order of is not 2. We need to show that is nonelementary. If is elementary, then (see [5], Section 5.1]). (i) is elementary of type I; each nontrivial element is elliptic. Since the order of is not or , neither is the order of . If then from Tables 1–3, is neither the finite spherical triangle groups and nor the dihedral groups and . If , then , and . On the other hand, by Lemma 7, , a contradiction.(ii)Suppose is an elementary group of type II; then, is conjugate to a subgroup of fixing whose every element is parabolic of the form , . Thus, the group is abelian, and hence, a contradiction to being Kleinian.(iii)Suppose is an elementary group of type III; then, both and are elliptic or both are loxodromic. In either case is abelian and as above this is a contradiction.We have shown that cannot be elementary if does not have order two. Hence, in all two cases, is a Kleinian group.
Notice that a Kleinian group with elliptic of order or and cannot have . If so, then and have a common fixed point. But this discrete group cannot be a Euclidean triangle group as they do not have elliptics of order or greater than . Thus, and have two common fixed points and the same trace. Hence, and must fix or interchange the fixed points of . The first is not possible, and the second shows the identity.
5. Projections and Principal Characters
Our methods to establish new inequalities rely on a fundamental result concerning spaces of finitely generated Kleinian groups. Namely they are closed in the topology of algebraic convergence—a result originally due to Jørgensen [7]. We state this now.
Definition 12. A sequence of -generator subgroups is said to be convergent algebraically to a -generator subgroup in if for each , the sequence of the corresponding generators converges uniformly to in the spherical metric of
Theorem 13. (Jørgensen).
That the map back is an isomorphism was proved by Jørgensen in [7]. There is a higher dimensional analogue to this Jørgensen’s theorem, but additional hypotheses are necessary—for instance giving a uniform bound on the torsion in a sequence or asking that the limit set be in geometric position (see [11], Proposition 5.8).
Now, we show the slice is closed. Notice that we are free to normalize a sequence of two-generator groups by conjugation and to pass to subsequences. First show a preliminary result.
Theorem 14. The slice
is closed in the two complex dimensional space in the usual topology.
Proof. Suppose that is a sequence of principal characters of Kleinian groups, say . We proceed by considering two cases: up to subsequence, is parabolic for all or not.
Case (a). Suppose is parabolic for all . Conjugate each so that the first generator is now represented by the matrix . Then, is constant and it remains to show that the sequence also converges. Since , we suppose the matrix for second generator is
We calculate that , and hence, . Jørgensen’s inequality gives for all , and hence, .
Set
Since commutes with is left unchanged under conjugation. The result follows by Theorem 13.
Case (b). We now suppose that is not parabolic for all . By conjugation, we may assume that is represented by the matrix
where Thus, the parameter which gives the quadratic equation
and hence, Since converges to Noting that for either choice of , we conclude that converges to the element represented by the matrix
It remains to show that the order two elements also converge. Note first that and so we may assume that . As in the previous case, the matrix for can be written as
so that and . Consider the conjugacy of the group by the following diagonal matrix,
and then, (replaced by the conjugacy ) has the form
Since commutes with is left unchanged under the conjugation.
The commutator is now
Hence, the parameter for the group is
At this point we would like to show , but this can actually happen in limits of Dehn surgeries—see the following example. Thus, we have two subcases:
Case (b) (i) Suppose
Solving for yields
Let then converges to the element of represented by the matrix
and . Applying Theorem 13, the limit group is Kleinian.
Case (b) (ii) Suppose . This is equivalent to . For each , we may conjugate and so that has a fixed point at and has a fixed point at . Thus, is upper triangular and is lower triangular. We may further conjugate by a diagonal matrix to achieve the upper entry in the matrix representing is . With this normalization we have
and . We calculate that
It follows that , for one choice of sign. Thus, and This group is nonelementary by Theorem 13 so . One can see this since as soon as , Jørgensen’s inequality gives . This completes the proof.
Example 1. We next give an example to show the last case can occur. That is and (and ). Let Then, is (a representation of) the two bridge figure of eight knot group if . The relator in this group is If we perform Dehn surgery on the complement of this knot, we obtain a Kleinian group generated by two elliptics of order , , and . Then, up to conjugacy and have the form Then, the relator (47) holds if and only if or its conjugate. Thus, is the group obtained from this orbifold Dehn surgery. As such it is discrete and nonelementary. Then, as , As , we find an involution such that with .
We next have the following lemma.
Lemma 15. Let be a sequence of principal characters of the two-generator Kleinian groups and let be the principal character of two-generator group Suppose that converges to and is not elliptic of order . Then, and
Proof. Since is not elliptic of order , is likewise not elliptic of these orders for all but finitely many . Applying Theorem 11, is a sequence of Kleinian groups with corresponding principal characters which converge to Now, is a Kleinian group generated by two elements of the same trace. Thus, , and hence, and
Now we may apply a result by Cao [12, Theorem 5.1] which gives a universal lower bound on the commutator parameter for such groups.
Thus, It follows that and
Theorem 16. The two complex dimensional space
is closed in the two complex dimensional space in the usual topology.
Proof. Suppose that is a sequence in with limit in By the definition of there is a sequence so that is the principal character of a Kleinian group. Then, we project to the principal character of . If infinitely many of these groups are Kleinian, the result we seek follows from Theorem 14. After passing to a subsequence, we are left with the case is elementary for every .
Case (i). Suppose is not elliptic of order for infinitely many . Then, Lemma 15 tells us that , and hence, is Kleinian, so this case cannot occur
Case (ii). We may now suppose, after passing to a subsequence, that each is elliptic of order , , . is a discrete elementary group generated by elliptics of orders and 2. The order of any such group is less than that of . If infinitely many of these groups are finite, then in fact is elliptic of order , or . Thus, lies in a finite set of values. Again, after a subsequence, we have . Then, and the latter group is Kleinian by hypothesis. Thus,
Case (iii). We are left with the situation that is generated by elliptics of orders and 2, infinite and elementary for each . The classification tells us this only happens for the Euclidean triangle groups and any such group has a common fixed point. Then, which is not possible as is a Kleinian group.
Data Availability
The manuscript includes the underlying data supporting the results of our study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by UAEU UPAR Grant (31S315). This paper is partly contained in the first author’s Ph.D. thesis.