# Spherical CR Geometry and Dehn Surgery (AM-165)

Richard Evan Schwartz
Pages: 200
https://www.jstor.org/stable/j.ctt6wq0j8

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xii)
4. PART 1. BASIC MATERIAL
• Chapter One Introduction
(pp. 3-11)

Dehn fillingis a basic surgery one can perform on a 3-manifold. LetMbe a 3-manifold that is the interior of a compact manifold with boundary\overline{M}. We say that atorus endofMis a torus boundary component of\overline{M}. LetEbe such a torus end, and letα\inH1(E) be a primitive homology element; i.e.,αis not a multiple of anotherβ\inH1(E). Let Σ be a solid torus with boundary ∂Σ. Letf:E→ ∂Σ be a homeomorphism such thatf*(α) = 0 inH1(Σ). Then the identification space...

• Chapter Two Rank-One Geometry
(pp. 12-22)

The reader may wish to consult [B] and [R], which are good references for real hyperbolic geometry. LetH^{n}denoten-dimensional hyperbolic space. There are several natural models forH^{n}, and we will discuss three of these.

Klein Model: In theKlein model,H^{n}is the open unit ball inR^{n}, which in turn is considered an affine patch of real projective spaceRP^{n}. In this model the isometries ofH^{n}are given by projective (i.e., line–preserving) transformations, which stabilize the open unit ball. In the invariant Riemannian metric, the geodesics are Euclidean line segments.

Poincaré...

• Chapter Three Topological Generalities
(pp. 23-31)

IfXis a metric space, then we can equip the set of compact subsets ofXwith theHausdorff metric. The distance between two compactK_{1},K_{2}\subset Xis defined as the infimalεsuch thatK_{j}is contained in theε-tubular neighborhood ofK_{3-j}forj= 1, 2. This metric induces theHausdorff topologyon closed subsets ofX. A sequence\left \{ S_{n} \right \}of closed subsets converges toSif, for every compactK\; \subset \; X, the Hausdorff distance betweenS_{n}\; \cap \; KandS\; \cap\; Kconverges to 0 asn→ ∞.

One of the cases of interest to us is the case...

• Chapter Four Reflection Triangle Groups
(pp. 32-40)

In this chapter we elaborate on the discussion in Section 1.4. Here is a well-known result from real hyperbolic geometry.

Lemma 4.1Letζ = (ζ0, ζ1, ζ2)be a triple of positive numbers such that\sum \zeta _{i}^{-1}< 1.Then there exists a geodesic triangleT_{\zeta }\; \subset \; H^{2}whose angles are\pi /\zeta _{0},\; \pi /\zeta _{1},and\pi /\zeta _{2}.This triangle is unique up to isometry.

Proof: Let\theta_{i}=\pi /\zeta _{i}. The easiest way to see the existence is as follows: Start with two linesL1andL2through the origin in the disk model that make an angle ofθ0. Now take a lineL0that makes...

• Chapter Five Heuristic Discussion of Geometric Filling
(pp. 41-50)

The proof of Theorem 1.1, at least as outlined in [T0], breaks down into the following three parts.

LetMbe a cusped hyperbolic 3-manifold. We have a discrete embedding\rho:\pi _{1}(M)\rightarrow \mathrm{Isom}(H^{3})so thatM = H^{3}/\Gamma, where Γ = π(M). If one perturbs the structure onM, then the resulting structure is determined by theholonomy representation\hat{\rho }:\pi^{_{1}}\left ( M \right )\rightarrow \mathrm{Isom}\left ( H^{^{3}} \right ).

Let\hat{\Gamma } = \hat{\rho }\left ( \pi _{1}\left ( M \right ) \right ). Suppose\hat{\rho }maps theZ² subgroups of π1(M) to discrete loxodromicZsubgroups of Isom(H³). ThenH^{^{3}}/\widehat{\Gamma}is a closed hyperbolic 3-manifold obtained by Dehn filling the cusp ofM.

There are enough perturbations of the hyperbolic structure onM...

5. PART 2. PROOF OF THE HST
• Chapter Six Extending Horotube Functions
(pp. 53-55)

LetPbe a parabolic element, fixing a pointp\in S^{3}. Letf:S^{3} - p\rightarrow Rbe a horotube function, as in Section 2.6.2. We assume thatf\left ( S^{3}-p \right )\subset \left [ 0,3 \right ]. In particular,fis bounded. In this chapter, we will explain how we extendftoCH² (and a bit outsideCH² as well). The simplest way to extendfwould be to imitate the way that a function on the circle is extended to a harmonic function on the unit disk. We could integrate against aPU(2, 1)-invariant kernel. This method is simple and intuitive but leaves us with a function that is...

• Chapter Seven Transplanting Horotube Functions
(pp. 56-60)

Letp \in PU\left ( 2,1 \right )be a parabolic element, fixingp. LetUpbe as in Equation 6.1. We use the Siegel model from Section 2.3, withp = \inftyandUp=C². Let {Pn} be a sequence of elements inPU(2, 1) with the property that thatPnPgeometrically, in the sense of Section 3.1.

Lemma 7.1 (Transplant)Suppose that the sequence {Pn} converges geometrically to P. Let F : Up→ R be a P-invariant smooth function. For each n there is a Pn-invariant smooth function Fn: Up→ R. Moreover, {Fn} → F inC^{\infty}\left ( U_{p} \right ).

For ease of...

• Chapter Eight The Local Surgery Formula
(pp. 61-65)

Recall from Section 2.6.2 that a horotube isniceif its boundary is a smooth cylinder and the horotube is stabilized by a 1-parameter parabolic subgroup. In this chapter we will describe how the Dehn surgery works for individual nice horotubes. In Chapter 13 we will see this picture occur around each cusp of the manifoldM= Ω/Γ from the HST.

LetTbe a nice horotube, stabilized by a parabolic group\left \langle P \right \rangle. We normalize so thatPis either as in Equation 2.12 or Equation 2.13. In the former case we also assume thatu≠ −1....

• Chapter Nine Horotube Assignments
(pp. 66-71)

In the first three chapters of Part 2 we have considered individual horotubes and horotube functions. Now we thicken the plot and consider collections of these objects. In this chapter we do not assume that our group Γ is a horotube group. However, we assume that Γ shares many properties of a horotube group, namely the following.

Γ is discrete.

The limit set Λ is more than a single point.

The regular set Ω is nonempty.

Γ has a finite and nonzero number of conjugacy classes of rank one parabolic subgroups, and no rank-2 parabolic subgroups.

It would seem reasonable...

• Chapter Ten Constructing the Boundary Complex
(pp. 72-77)

Henceforth we assume that Γ is a horotube group. Our first goal is to prove the following lemma.

Lemma 10.1 (Structure)Let E1,...,Enbe the horocusps ofΩ/Γ.There are horotubes\tilde{E}_{1},\ldots,\tilde{E_{n}}and elements\gamma _{1}, \ldots,\gamma _{n}\in \Gammasuch thatE_{j}=\tilde{E}_{j}/\left \langle \gamma _{j} \right \rangle.Furthermore every parabolic element ofΓis conjugate to a power of someγj.Thus, any maximal Z parabolic subgroup ofΓis conjugate inΓto some\left \langle \gamma _{j} \right \rangle.

Let Σ be the good spine for M = Ω/Γ guaranteed by Lemma 3.2. Then the horocuspsE1,...,Encan be taken as the components of M – Σ. Let\Psi ^{\infty }be the lift to Ω...

• Chapter Eleven Extending to the Inside
(pp. 78-84)

In this chapter we will assume for ease of exposition that Γ is a horotube group with no elliptic elements. At the end of the proof of the HST in Chapter 13, we will discuss how to modify our arguments when there exist elliptic elements.

We have already constructed a horotube function assignmentpfp, which defines the complex\Psi ^{\infty }. In this chapter we will extend the functionfpso that the extensionFpis smooth on the setUpfrom Chapter 6 and has the kind of transversality guaranteed by Equation 3.3, at least for “relevant” sets...

• Chapter Twelve Machinery for Proving Discreteness
(pp. 85-90)

The method we present bears some resemblance to the Poincaré fundamental polyhedron theorem (see [FZ], [M1]), but is really a reformulation of Thurston’s holonomy theorem (see [T0], [CEG]).

In Section 12.2, we will define the notion of asimple complex. This is just a higher dimensional version of the concept discussed in [Mat] for 3-manifolds. The complexes Ψ and\Psi ^{\infty }are both simple complexes.

In Section 12.3, we will use simple complexes to define the basic object that we call a (G,X)-chunk. Here (G,X) refers to the data for a geometric structure. Intuitively, (G,X)-chunks are the pieces one gets after...

• Chapter Thirteen Proof of the HST
(pp. 91-102)

In this chapter we will assemble the ingredients from the previous chapters to prove the Horotube Surgery Theorem (HST). Our proof has two themes. One theme is essentially a detailed working out of the material in Sections 5.5–5.6. The other theme is a discreteness proof using the machinery from Chapter 12. Until Section 13.9 we make the blanket assumption that Γ has no elliptic elements. At the end of this chapter we will explain how to deal with elliptic elements.

We will use the notation from the statement of the HST. As in Chapter 5, we will use the...

6. PART 3. THE APPLICATIONS
• Chapter Fourteen The Convergence Lemmas
(pp. 105-112)

In this chapter we investigate the interplay between the ordinary convergence of sequence of elements {Pn} ЄPU(2,1) to somePЄPU(2,1) and the stronger geometric convergence defined in Section 3.1. Recall that a parabolic element isR-parabolic iff it stabilizes anR-slice and isC-parabolic iff it stabilizes a uniqueC-slice. In theR-parabolic case, the map is conjugate to the map in Equation 2.13 withs= 0. In theC-parabolic case, the map is conjugate to the one in Equation 2.12 withu≠ 1. We say that a parabolic elementPisirregularif it...

• Chapter Fifteen Cusp Flexibility
(pp. 113-120)

We want to apply the HST to the group Γ3from Theorem 1.4, so we need some perturbations. In this chapter we construct the basic examples.

Let\cal {R}denote the space of representations ofZ3*Z3intoSU(2, 1) in which the two generators map to order-3 elliptics acting freely onS³. As is typical, we consider two representations the same if they are conjugate inSU(2, 1). The dimension count we give below shows that\cal {R}is a 4-dimensional analytic manifold, at least in the region of interest to us.

We say that adiamond groupis a representationx \in \cal {R}...

(pp. 121-123)

The goal of this chapter is to prove Theorem 1.5.

We fix a left-invariant Riemannian metric onSU(2, 1) just for the convenience of being able to give names to open sets. The choice of metric doesn’t really matter. LetP\subset SU\left ( 2,1 \right )be an element, and letBє(P) be the ϵ-ball aboutP. We say that atrace neighborhoodofPis a subsetY\subset SU\left ( 2,1 \right )such thatP\in Y, and for any ϵ > 0 the set

X_{\epsilon }:= \{ \textrm{Tr}(h)|h \in Y \cap B_{\epsilon }(P) \}(16.1)

contains an open neighborhood of Tr(P) inC.

Here is an obvious consequence of this definition.

Lemma 16.1Let Y be a trace...

(pp. 124-130)

In this chapter we will prove Theorem 1.7. Our first goal is to construct a finite index subgroup of Γ3based on a finite trivalent treeY. There is a tiling\cal{T}ofH² by ideal triangles, that is invariant under the action of the groupG3considered in the previous chapter. Indeed, ifτis a single ideal triangle, then the tiling is created by considering the orbit ofτunder the ideal triangle group generated by reflections in the sides ofτ.

Given any ideal polygonIinH² that is tiled by a finite number of...

• Chapter Eighteen Small-Angle Triangle Groups
(pp. 131-136)

The goal of this chapter is to prove Theorem 1.10. We will use the notation from Chapter 4. Recall that ◊Rep(ζ) consists of those complex hyperbolic ζ-triangle group representations such that the productI0I1I2is loxodromic. First, we will pin down the structure of ◊Rep(ζ). Let α(∞) denote the parameter in ◊Rep(∞,∞,∞) corresponding to the golden triangle group.

Lemma 18.1If |ζ| is sufficiently large, then◊Rep(ζ)is a proper subinterval ofRep(ζ),and its critical endpoint α(ζ)converges to α(∞).

Proof: Ifαis much larger thanα(∞) and min |ζ| is large, thenI0I1I2is loxodromic by continuity....

7. PART 4. STRUCTURE OF IDEAL TRIANGLE GROUPS
• Chapter Nineteen Some Spherical CR Geometry
(pp. 139-143)

This chapter‘ begins Part 4 of the monograph. In this part we prove Theorems 1.4 and 1.3.

We introduced thehybrid coneconstruction in [S1] and used it in [S0] to definehybrid spheres. In [FP], hybrid spheres were renamedR-spheres. To be consistent, we rename the hybrid cone theR-cone.

ForFє {C,R} recall that anF-circleis the accumulation set, inS³, of anF-slice. Say that anF-arcis a connected arc of anF-circle. Given a curveC\subset S^{3}and a pointqєS³ −C, we want to produce a surface by coningC...

• Chapter Twenty The Golden Triangle Group
(pp. 144-155)

The goal of this chapter is to prove everything in Theorem 1.4 except the statement about Ω/Γ3. We will deal with this last statement in Chapter 21.

LetI0,I1,I2be the generators of Γ’, the golden triangle group—see Equation 4.13. The elementIjfixes pointwise aC-circleCj. TheC-circlesCi−1andCi+1intersect pairwise in a pointvi. The elementIiIjIkfixes a pointqjand stabilizes aC-circleEj. SinceIkIjIiandIiIjIkare inverses of each other, they both stabilize the flag (Ej,qj). A routine calculation shows thatCilinks...

• Chapter Twenty One The Manifold at Infinity
(pp. 156-164)

In this chapter we will prove the last statement of Theorem 1.4—Ω/Γ3is homeomorphic to the Whitehead link complement. We already have a fundamental domain for the action of Γ, namely,F\cup I_{0}\left ( F \right ), whereFis as in Equation 20.3. We really just have to analyze the topology of this domain and its side pairings. This is the approach taken in [S0]. Here we will take a more combinatorial (and long-winded) approach because it gives us a global understanding of the way Γ3acts on Ω. Also, the combinatorial approach shows off some of the beauty of Γ3.

In...

• Chapter Twenty Two The Groups near the Critical Value
(pp. 165-175)

In this chapter we prove Theorem 1.3 for parameter valuess\in\left ( \underline{s}-\epsilon,\bar{s} \right ). Here\bar{s}=\sqrt{125/3}is the critical parameter,\underline{s}=\sqrt{35}is the “Goldman-Parker parameter”, and ϵ is some small but unspecified value. As we mentioned in Chapter 1, our proof relies on some technical details from [S5].

The constructions we make here are “loxodromic analogues” of the constructions we made in Chapters 18–19, which were based on the parabolic elementI1I0I2. Here will explain enough about these constructions at least to show two computer plots that give evidence for the main result, which is proved in [S5].

This section parallels...

• Chapter Twenty Three The Groups far from the Critical Value
(pp. 176-180)

In [GP] Goldman and Parker proved that the complex hyperbolic ideal triangle group Γ = ρs(G) is discrete, provided thats\in \left [ 0,\underline{s} \right ]. A certain transition occuring at the parameter\underline{s}made it impossible for their method to work in(\underline{s},\bar{s}]. The transition at\underline{s}is invisible to our proof given in the previous chapter. However, our proof in the previous chapter becomes much more difficult for parameters much less than\underline{s}for other technical reasons. For this reason, we proved Theorem 1.3 for parameters in\left ( \underline{s}-\epsilon,\bar{s} \right ), avoiding very small parameters but still covering the somewhat tricky parameter\underline{s}...

8. Bibliography
(pp. 181-184)
9. Index
(pp. 185-186)