A complex hyperbolic Riley slice
Summary (5 min read)
1.1 Context and motivation
- This question has been addressed in many different contexts.
- When is discrete, such deformations are called quasi-Fuchsian.
1.2 Main definitions and discreteness result
- The direct analogue of the Riley slice in complex hyperbolic plane would be the set of conjugacy classes of groups generated by two noncommuting, unipotent parabolic elements A and B of SU.2; 1/. (Note that in contrast to PSL.2;C/, there exist parabolic elements in SU.2; 1/ that are not unipotent.
- The authors shall also indicate various other interesting discrete groups in U but these will not be their main focus.
- 1Parker has one of Riley’s printouts of this picture dated 26th March 1979.
- When ˛1 and ˛2 are both zero, that is, at the origin of the square, the group hA;Bi is R–Fuchsian.
- The quotient of the Lagrangian plane preserved by hA;Bi is a hyperbolic thrice-punctured sphere where the three (homotopy classes of) peripheral elements are represented by (the conjugacy classes of) A, B and AB .
1.3 Decompositions and triangle groups
- Deformations of triangle groups in PU.2; 1/ have been considered in many places, including Goldman and Parker [16], Parker, Wang and Xie [25], Pratoussevitch [28] and Schwartz [32].
- Groups generated by complex reflections of higher order are also interesting; see Mostow [22] for example, but the authors do not consider them here.
- Schwartz [32] develops a series of conjectures about which points in this space yield discrete and faithful representations of the triangle group.
- The gap between the vertical segment in Figure 1 and the curve where ŒA;B is parabolic illustrates the nonoptimality of the result of [37].
1.4 Spherical CR uniformisations of the Whitehead link complement
- Its boundary at infinity is a circle bundle over the surface.
- A hyperbolic manifold arising in this way was first constructed by Schwartz: Geometry & Topology, Volume 21 (2017) A complex hyperbolic Riley slice 3397 Theorem 1.4 [30].
- Deraux [7] proved that this uniformisation was flexible: he described a one-parameter deformation of the uniformisation described by Deraux and Falbel [8], each group in the deformation being a uniformisation of the figure-eight knot complement.
- The representation of the Whitehead link group the authors consider here was identified from a different point of view by Falbel, Koseleff and Rouillier [13, page 254] in their census of PGL.3;C/ representations of knot and link complement groups.
- These groups have Cartan invariant AD˙ arccos p 3=8.
1.5 Ideas of the proofs
- The Ford domain is invariant by the subgroup generated by A and the authors obtain a fundamental domain for the group by intersecting the Ford domain with a fundamental domain for the subgroup generated by A.
- The authors believe that it should be possible to mimic Riley’s approach and to construct regions in their parameter space where the Ford domain is more complicated.
- As with Riley’s work, this may only be accessible via computer experiments.
- In order to prove this result, the authors analyse in detail their fundamental domain, and show that it gives the classical description of the Whitehead link complement from an ideal octahedron equipped with face identifications.
1.6 Further remarks
- Other discrete groups appearing in U As well as the ideal triangle groups and bending groups discussed above, there are some other previously studied discrete groups in this family.
- There is, conjecturally, one extremely significant difference between the classical Riley slice and their complex hyperbolic version.
- This curve provides a one-parameter family of (conjecturally discrete) representations that connects Schwartz’s uniformisation of the Whitehead link complement to ours.
- Possibly, this is because deformations of fundamental domains with tangencies between bisectors are complicated.
1.7 Organisation of the article
- In Section 2 the authors present the necessary background facts on complex hyperbolic space and its isometries.
- The authors state and apply the Poincaré polyhedron theorem in Section 5.
- The authors would like to thank Miguel Acosta, Martin Deraux, Elisha Falbel and Antonin Guilloux for numerous interesting discussions.
- Will thanks Craig Hodgson, Neil Hoffman and Chris Leininger for kindly answering his naïve questions.
- Will had the pleasure to share a very useful discussion with Lucien Guillou on a sunny June afternoon in Grenoble.
2.1 The complex hyperbolic plane
- The standard reference for complex hyperbolic space is Goldman’s book [15].
- A lot of information can also be found in Chen and Greenberg’s paper [3]; see also the survey articles [24; 38].
- The complex hyperbolic plane is endowed with the Bergman metric ds2 D 4 hz; zi2 det hz; zi hdz; zi hz; dzi hdz; dzi : The Bergman metric is equivalent to the Bergman distance function defined by cosh2 .m; n/ 2 D hm;nihn;mi hm;mihn;ni ; where m and n are lifts of m and n to C3 .
- Horospherical coordinates give a model of complex hyperbolic space analogous to the upper half-plane model of the hyperbolic plane.
- The Cygan metric dCyg on @H 2C fq1g plays the role of the Euclidean metric on the upper half-plane.
2.2 Isometries
- Since the Bergman metric and distance function are both given solely in terms of the Hermitian form, any unitary matrix preserving this form is an isometry.
- Define U.2; 1/ to be the group of unitary matrices preserving the Hermitian form and PU.2; 1/ to be the projective unitary group obtained by identifying nonzero scalar multiples of matrices in U.2; 1/.
- It is Geometry & Topology, Volume 21 (2017) A complex hyperbolic Riley slice 3403 called elliptic when it has (at least) one fixed point inside H 2C .
- The following criterion distinguishes the different isometry types: Proposition 2.3 [15, Theorem 6.2.4].
- This implies that T is upper triangular with each diagonal element equal to 1. Lemma 2.5 [15, Section 4.2].
2.3 Totally geodesic subspaces
- Maximal totally geodesic subspaces of H 2C have real dimension 2, and they fall into two types.
- A more general description of cross-products in Hermitian vector spaces can be found in [15, Section 2.2.7].
- The other type of maximal totally geodesic subspace is a Lagrangian plane.
- In particular, real planes are fixed points sets of antiholomorphic isometric involutions (sometimes called real symmetries).
2.4 Isometric spheres
- Using the characterisation (4) of the Cygan metric in terms of the Hermitian form, the following lemma is obvious: Lemma 2.9 Suppose that B 2 SU.2; 1/ written in the form (7) does not fix q1 .
- Therefore their method will be to guess the Ford polyhedron and check this using the Poincaré polyhedron theorem.
2.5 Cygan spheres and geographical coordinates
- The authors now give some geometrical results about Cygan spheres.
- They are, in particular, applicable to isometric spheres.
- The Cygan sphere SŒw;s .r/ of radius r with centre Œw; s is the image of SŒ0;0 .r/ under the Heisenberg translation TŒw;s .
- The intersection of two Cygan spheres is connected.
3.1 Coordinates
- In fact, a slightly stronger statement will follow from Theorem 3.2 below.
- The conditions A.pBA/D pAB and B.pAB/D pBA imply that there exist two nonzero complex numbers and satisfying ApBA D pAB and BpAB D pBA: Indeed, once this quadruple is fixed, A and B are uniquely determined by Proposition 2.6.
- Provided that ST and TS are unipotent, this suffices to prove the second item by Proposition 2.6.
- The authors will use the notation S and T for these order-three symmetries throughout the paper.
- This follows directly from the following: Proposition 3.4 Proof.
3.3 Symmetries of the moduli space
- There is an antiholomorphic involution with the properties: (1) interchanges pA and pB and interchanges pAB and pBA .
- Hence all three conditions in the first part are equivalent.
- The properties of R– and C–decomposability have also been studied (in the special case of pairs of loxodromic isometries) from the point of view of traces in SU.2; 1/ in [36], and (in the general case) using cross-ratios in [27].
3.4 Isometry type of the commutator
- The isometry type of the commutator will play an important role in the rest of this paper.
- It is easily described using the order-three elliptic maps given by Proposition 3.3.
- This would mean that A and B commute, which cannot be because their fixed point sets are disjoint.
4.1 Isometric spheres for S , S 1 and their A–translates
- The polyhedron D is their guess for the Ford polyhedron of , subject to the combinatorial restriction discussed in Section 4.2.
- The authors start with the isometric spheres I.S/ and I.S 1/ for S and its inverse.
- The authors now fix some notation: Geometry & Topology, Volume 21 (2017) 3416 John R Parker and Pierre Will Definition 4.1 Similarly, the isometric sphere I k has radius 1 and centre the point with Heisenberg coordinates Œk`AC p cos˛1ei˛2 ; sin˛1 .
- This gives the part about centres by a straightforward verification.
4.2 A combinatorial restriction
- The following section is the crucial technical part of their work.
- It relies on Proposition 4.6, describing the set of points where D.x4 1 ;x4 2 / > 0, and on Proposition 4.7, which gives geometric properties of the triple intersection.
- The following fact will be crucial in their study; compare Figure 5.
- Then the isometric sphere IC0 is contained in the exterior of the isometric spheres.
- The proof of Proposition 4.8 will be detailed in Section 7.4.
5.2 Application to our examples
- The authors are now going to apply Theorem 5.1 to the group generated by S and A. Explicit matrices for these transformations are provided in (13) and (16).
- The authors can now verify that satisfies the first condition of being a side pairing.
- The intersection of D with UC 0 is the same as the intersection of UC 0 with the Ford domain DS for the order-three group hSi.
- Thus the Poincaré polyhedron theorem gives the presentation (19).
- This follows from the fact (Chuckrow’s theorem) that the algebraic limit of a sequence of discrete and faithful representations of a nonvirtually nilpotent group in Isom(H nC ) is discrete and faithful (see for instance [4, Theorem 2.7] or [21] for a more general result in the framework of negatively curved groups).
6 The limit group
- This polyhedron is slightly more complicated than the one in the previous section due to the appearance of ideal vertices that are the points in VS 1T and VST 1 .
- Geometry & Topology, Volume 21 (2017) 3426 John R Parker and Pierre Will (2) We analyse the combinatorics of the ideal boundary @1D of this polyhedron.the authors.the authors.
- More precisely, the authors will see that the quotient of @1Dn.
6.2 The Poincaré theorem for the limit group
- The limit group has extra parabolic elements.
- Therefore, in order to apply the Poincaré theorem, the authors must construct a system of consistent horoballs at these parabolic fixed points (see Section 5.1).
- Lemma 6.1 Proof Projecting vertically — see Remark 2.13 — the authors see that the projections of I 1 and IC 1 are tangent discs and, as they are strictly convex, their intersection contains at most one point.
- The other tangency is along the same lines.
6.3 The boundary of the limit orbifold
- The manifold at infinity of the group lim is homeomorphic to the Whitehead link complement.
- Recall that the (ideal boundary of) the side s˙ k is the part of @I˙ k which is outside (the ideal boundary of) all other isometric spheres.
- As a consequence, the two connected components of the common exterior are either exchanged or both preserved.
- Applying powers of A gives the other quadrilaterals and bigons.
- The following result, which will be proved in Section 7.5, is crucial for proving Theorem 6.4.
7.2 The region Z is an open disc in the region L: proof of Proposition 4.6
- Likewise, it is negative in the part below the zero locus, that is, containing the point .x;y/D 3; 3 2 .
- A complex hyperbolic Riley slice 3439 continuous bijection connecting the points .
7.3 Condition for no triple intersections: proof of Proposition 4.5
- The numerical condition given in the statement of Proposition 4.5 will follow from the next lemma.
- This implies that it is not possible to pass from one of the above types to another without passing through a polynomial having a double root.
- The authors saw in the second part of Lemma 7.7 that on both these intervals L˛1;˛2.T / has no real roots, that is its roots are of type (a).
- This completes the proof of Proposition 4.5.
7.4 Pairwise intersection: proof of Proposition 4.8
- Proposition 4.8 will follow from the next lemma.
- To prove the disjointness of the given isometric spheres the authors calculate the Cygan distance between their centres.
- This will not be the case in their examples.).
- The authors will show that dCyg. Ak. A complex hyperbolic Riley slice 3443 7.5.
- The part about intersections of fans and isometric spheres is proved easily by projecting vertically onto C , as in the proof of Proposition 4.8 .
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Cites background from "A complex hyperbolic Riley slice"
...spherical CR structure on a closed hyperbolic 3 manifold. Recently, in [15] Deraux and Falbel have described a spherical CR structure on the complement of the figure eight knot. In the article to come [87], Parker and Will produce an example of a spherical CR structure on the complement of the Whitehead link that is not conjugate to Schwartz’s one. The question of knowing which hyperbolic 3 manifolds a...
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References
87 citations
"A complex hyperbolic Riley slice" refers background in this paper
...In [32], Schwartz develops a series of conjectures about which points in this space yield discrete and faithful representations of the triangle group....
[...]
...Deformations of triangle groups in PU(2, 1) have been considered in many places, among which ([16, 28, 32, 26])....
[...]
...1 of [32] states that a complex hyperbolic...
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82 citations
"A complex hyperbolic Riley slice" refers background in this paper
...The title of this article refers to the so-called Riley slice of Schottky space (see [19; 1])....
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...The boundary of the classical Riley slice is not a smooth curve and has a dense set of points where particular group elements are parabolic (see for instance the beautiful picture in the introduction of Keen and Series [19])....
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...He was particularly interested in the (closed) set where is discrete and free, which is now called the Riley slice of Schottky space; see Keen and Series [19]....
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Frequently Asked Questions (3)
Q2. What is the image of the holonomy representation of the Whitehead link complement?
It should be noted that the image of the holonomy representation of their uniformisation of the Whitehead link complement is the group generated by S and T , which is isomorphic to Z3 Z3 .
Q3. How do the authors prove that the spheres are disjoint?
Since all the isometric spheres have radius 1, if the authors can show their centres are a Cygan distance at least 2 apart, then the spheres are disjoint.