j. differential geometry

51 (1999) 471-515

HAUSDORFF DIMENSION AND CONFORMAL DYNAMICS I:

STRONG CONVERGENCE OF KLEINIAN GROUPS

CURTIS T. MCMULLEN

Abstract

This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λ_{n} and Λ of a sequence of Kleinian groups Γ_{n} → Γ, where M = ℍ^{3}/Γ is geometrically finite. We show if Γ_{n} → Γ strongly, then:

(a) M_{n} = ℍ^{3}/Γ_{n} is geometrically finite for all n ≫ 0,

(b) Λ_{n} → Λ in the Hausdorff topology, and

(c) H. dim(Λ_{n}) → H. dim(Λ), if H. dim(Λ) ≥ 1.

On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H. dim(Λ) < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.

1. Introduction

To any complete hyperbolic manifold M one may associate a conformal dynamical system, by considering the action of Γ = π_{1}(M) on the sphere at infinity for the universal cover, S_{∞}^{d} = ∂ℍ^{d+1}. For 3manifolds one obtains in this way the classical Kleinian groups acting on the Riemann sphere ℂ̂.

Received October 3, 1997, and, in revised form, October 17 1998. Research partially supported by the NSF.

1991 Mathematics Subject Classification: Primary 30F40; Secondary 58F11, 58F23.

A fundamental invariant of M and Γ is the Hausdorff dimension of the limit set Λ ⊂ S_{∞}^{d}, the set of accumulation points of any orbit Γx ⊂ ℍ^{d+1}. When Γ is geometrically finite, D = H. dim(Λ) coincides with several other invariants of Γ, and is related to the bottom of the spectrum of the Laplacian on M by

λ_{0}(M) = D(d − D),

when D ≥ d/2.

Moreover the limit set of a geometrically finite group supports a canonical conformal density of dimension D. That is, there is unique probability measure μ on Λ transforming by |γ′|^{D} under the action of Γ. In the absence of cusps, μ is simply the normalized D-dimensional Hausdorff measure on Λ.

In this paper we study the behavior of H. dim(Λ) for a sequence of Kleinian groups. Recall that Γ_{n} → Γ geometrically if the groups converge in the Hausdorff topology on closed subsets of Isom(ℍ^{d+1}). We say Γ_{n} → Γ strongly if, in addition, there are surjective homomorphisms

χ_{n} : Γ → Γ_{n}

converging pointwise to the identity. Equivalently, χ_{n} tends to the inclusion Γ ⊂ Isom(ℍ^{d+1}).

It has been conjectured that the Hausdorff dimension of the limit set varies continuously under strong limits (see e.g. [33, Conj. 5.6]). We will show this conjecture is false in general, and at the same time give positive theorems to guarantee continuity.

In the 3-dimensional case we obtain the following results.

Theorem 1.1. Let Γ_{n} → Γ strongly, where M = ℍ^{3}/Γ is geometrically finite. Then Γ_{n} is geometrically finite for all n ≫ 0, and the limits sets satisfy Λ_{n} → Λ in the Hausdorff topology.

Theorem 1.2. If, in addition, H. dim(Λ) ≥ 1, then

H. dim(Λ_{n}) → H. dim(Λ);

and if H. dim(Λ) > 1 then the canonical densities satisfy μ_{n} → μ in the weak topology on measures.

On the other hand we find new phenomena when H. dim(Λ) < 1:

Theorem 1.3. For any ϵ with 0 < ϵ < 1/2, there exist geometrically finite groups Γ and Γ_{n}, such that Γ_{n} → Γ strongly but

H. dim(Λ_{n}) → 1 > H. dim(Λ) = 1/2 + ϵ.

To recover continuity of dimension in general, one must sharpen the notion of convergence. Recall that Γ_{n} → Γ algebraically if there are isomorphisms χ_{n} : Γ → Γ_{n} converging to the identity. A parabolic element g ∈ Γ is an accidental parabolic if χ_{n}(g) is hyperbolic for infinitely many n. Then the complex length L_{n} + iθ_{n} of χ_{n}(g) tends to zero; it does so radially if θ_{n} = O(L_{n}), and horocyclically if θ_{n}^{2}/L_{n} → 0.

Theorem 1.4. Let M = ℍ^{3}/Γ be geometrically finite, and suppose Γ_{n} → Γ algebraically. Then:

(a) Γ_{n} → Γ strongly ⟺ all accidental parabolics converge horocyclically; and

(b) H. dim(Λ_{n}) → H. dim(Λ) and μ_{n} → μ if all accidental parabolics converge radially.

The examples of Theorem 1.3 reside in the gap between horocyclic and radial convergence.

Higher dimensional manifolds. Theorems 1.1 and 1.2 generalize to Kleinian groups acting on S_{∞}^{d}, d ≥ 3, if we replace H. dim(Λ) ≥ 1 by the condition

D = H. dim(Λ) ≥ (d − 1)/2.

Since λ_{0}(M) is sensitive to the dimension of the limit set only when D > d/2, we find:

Theorem 1.5. If M is a geometrically finite manifold of any dimension and M_{n} → M strongly, then λ_{0}(M_{n}) → λ_{0}(M).

Tame 3-manifolds. A hyperbolic manifold is topologically tame if it is homeomorphic to the interior of a compact manifold. If M = ℍ^{3}/Γ is geometrically infinite but topologically tame, then H. dim(Λ) = 2, and it is easy to see H. dim(Λ_{n}) → H. dim(Λ) whenever Γ_{n} → Γ geometrically, algebraically or strongly (§7).

On the other hand, the limit set of such a manifold generally carries many different conformal densities μ of dimension 2 (see §3), so the discussion of convergence of these measures must be reserved for the geometrically finite case.

Quasifuchsian groups. As applications of the results above, one can study a sequence of quasifuchsian manifolds M_{n} = Q(X_{n}, Y) in Bers’ model for the Teichmüller space of a surface S. Here are four examples, treated in detail in §9.

1. If X_{n} = τ^{n}(X_{0}), where τ is Dehn twist, then M_{n} has distinct algebraic and geometric limits M_{A} and M_{G} as n → ∞. We find M_{n} → M_{G} strongly and the limit sets satisfy

H. dim(Λ_{n}) → H. dim(Λ_{G}) > H. dim(Λ_{A}).

2. If X_{n} = ϕ^{n}(X_{0}), where ϕ is pseudo-Anosov on S or any subsurface of S, then all geometric limits are geometrically infinite and

H. dim(Λ_{n}) → 2.

3. If X_{n} is obtained by pinching a system of disjoint simple closed curves on X_{0} (in Fenchel-Nielsen coordinates), then M_{n} tends strongly to a geometrically finite manifold M and

H. dim(Λ_{n}) → H. dim(Λ) < 2.

4. Finally consider the manifolds M_{t} = Q(τ^{t}(X), Y), t ∈ ℝ, obtained by performing a Fenchel-Nielsen twist of length t about a simple geodesic C on X. In this case we show there is a continuous function δ(t), periodic under t ↦ t + ℓ_{C}(X), such that

lim_{t→∞} | H. dim(Λ_{t}) − δ(t)| = 0.

It seems likely that δ(t) is nonconstant, and thus H. dim(Λ_{t}) oscillates along this twist path to infinity in Teichmüller space.

This last example was inspired by a discovery of Douady, Sentenac and Zinsmeister in the dynamics of quadratic polynomials. These authors show the Hausdorff dimension of the Julia set J(z^{2} + c) is also asymptotically periodic, and probably oscillatory, as c ↘ 1/4 along the real axis [17].

Plan of the paper. In §2 and §3 we consolidate known material on the dimension of the limit set of a Kleinian groups. Some short proofs are included for ease of reference. The main results on continuity of dimension and λ_{0} are obtained in §7. Examples of discontinuity, including Theorem 1.3, are given in §8.

The general argument to establish continuity of dimension is to take a weak limit ν of the canonical densities μ_{n}, and show ν = μ. It turns out that μ ≠ ν only if ν is an atomic measure supported on cusps points in the limit set. To rule this out, we explicitly control the concentration of μ_{n} near incipient cusps. Part of the control comes from convergence of the convex core (§4), and part from estimates for the Poincaré series (§6). The theory of accidental parabolics is developed along the way (§5), and the applications to quasifuchsian groups are given in §9.

Notes and references. The continuity of H. dim(Λ) was also studied independently and contemporaneously by Canary and Taylor. Using spectral methods, Canary and Taylor show that if Γ_{n} → Γ strongly and M^{3} = ℍ^{3}/Γ is not a handlebody, then H. dim(Λ_{n}) → H. dim(Λ) [15]. The condition that M^{3} is not a handlebody guarantees H. dim(Λ) ≥ 1, so when M^{3} is geometrically finite their result is also covered by Theorem 1.2. On the other hand, the theorems of this paper do provide continuity of H. dim(Λ) for geometrically finite handlebodies, so long as a condition such as H. dim(Λ) > 1 or radial convergence Γ_{n} → Γ is assumed. Note that our counterexamples to continuity of dimension come from geometrically finite handlebodies with cusps (§8).

A version of Theorem 1.1 (without continuity of Λ) was proved by Taylor [33]. See Anderson and Canary for related results on cores and limits [2], [3].

This paper belongs to a three-part series [25], [24]. Part II gives parallel results in the setting of iterated rational maps. The theory of conformal densities is available for both rational maps and Kleinian groups; it is in anticipation of the applications in Part II that we work with conformal densities here, rather than with eigenfunctions of the Laplacian.

Part III presents explicit dimension calculations for families of conformal dynamical systems.

For background on hyperbolic manifolds, see the texts [34], [37], [6] and [28].

Notation. A ≍ B means A/C < B < CB for some implicit constant C; n ≫ 0 means for all n sufficiently large.

2. The basic invariants

Let ℍ^{d+1} be the hyperbolic space of constant curvature −1, and let S_{∞}^{d} = ℝ_{∞}^{d} ∪ {∞} denote its sphere at infinity. Let M = ℍ^{d+1}/Γ be a complete hyperbolic manifold. In this section we recall the relation between:

• λ_{0}(M), the bottom of the spectrum of the Laplacian;

• δ(Γ), the critical exponent of the Poincaré series for Γ;

• α(Γ), the minimum dimension of a Γ-invariant density; and

• H. dim(Λ_{rad}), the Hausdorff dimension of the radial limit set.

Groups and limit sets. A Kleinian group is a discrete subgroup Γ ⊂ Isom(ℍ^{d+1}). Every complete manifold of constant curvature −1 can be presented as a quotient M = ℍ^{d+1}/Γ where Γ is a Kleinian group.

For simplicity we will generally assume that Γ is torsion-free and nonelementary, i.e. any subgroup of finite index is nonabelian.

The limit set Λ of Γ is the subset of S_{∞}^{d} defined for any x ∈ ℍ^{d+1} by

Λ = \overline{Γx} ∩ S_{∞}^{d}.

It complement Ω = S_{∞}^{d} − Λ is the domain of discontinuity.

The radial limit set Λ_{rad} ⊂ Λ consists of those y ∈ S_{∞}^{d} such that there is a sequence γ_{n}x → y which remains within a bounded distance of a geodesic landing at y. Equivalently, y ∈ Λ_{rad} iff y corresponds to a recurrent geodesic on M.

Cusps. An element γ ∈ Γ is parabolic if it has a unique fixed-point c ∈ S_{∞}^{d}.

We say c ∈ S_{∞}^{d} is a cusp point, and its stabilizer L ⊂ Γ is a parabolic subgroup, if L contains a parabolic element. Then L contains a subgroup of finite index L_{0} ≅ ℤ^{r}, r > 0; and we say c and L belong to a cusp of rank r (compare [37, §4]). All cusp points belong to the limit set, and all elements of L − {id} are parabolic.

Invariants. The bottom of the spectrum of the Laplacian is defined by

(2.1) λ_{0}(M) = inf \{\frac{∫_{M} |∇f|^{2}}{∫_{M} |f|^{2}} : f ∈ C_{0}^{∞}(M)\}
(2.2)  = sup{λ ≥ 0 : ∃ f > 0 on M with Δf = λf}.

Here Δ denotes the positive Laplacian; for example λ_{0}(ℍ^{d+1}) = d^{2}/4. The equivalence of the two definitions above (on any Riemannian manifold) is shown in [16].

The Poincaré series is defined for x ∈ ℍ^{d+1} ∪ Ω by

P_{s}(Γ, x) = \{ \begin{matrix}∑_{γ∈Γ} e^{−sd(x,γx)} & if x ∈ ℍ^{d+1}, \\ ∑_{γ∈Γ} |γ′(x)|^{s} & if x ∈ Ω. \end{matrix}

Here and below |γ′| is measured in the spherical metric. The critical exponent is given by

δ(Γ) = inf{s ≥ 0 : P_{s}(Γ, x) < ∞};

it is independent of the choice of x.

A Γ-invariant conformal density of dimension α is a positive measure μ on S_{∞}^{d} such that

μ(γE) = ∫_{E} |γ′|^{α} dμ (2.3)

for every Borel set E and γ ∈ Γ. A density is normalized if μ(S_{∞}^{d}) = 1.

From a more functorial point of view, a conformal density of dimension α is a map

μ : (conformal metrics ρ(z)|dz| on S_{∞}^{d}) → (measures on S_{∞}^{d})

such that

\frac{dμ(ρ_{1})}{dμ(ρ_{2})} = (\frac{ρ_{1}}{ρ_{2}} )^{α} .

Conformal maps act on densities in a natural way, and (2.3) says γ_{∗}(μ) = μ. We implicitly identify μ with the measure μ(σ) where

σ = 2|dx|/(1 + |x|^{2})

is the spherical metric.

The critical dimension of Γ is given by

(2.4) α(Γ) = inf{α ≥ 0 : ∃ a Γ-invariant density of dimension α}.

In (2.2) and (2.4) it is easy to see that the inf and sup are achieved: there exist a positive eigenfunction with eigenvalue λ_{0}(M), and a Γinvariant density of dimension α(Γ). In particular we have α(Γ) > 0, because Λ has no Γ-invariant measure.

The Hausdorff dimension of the radial limit set, denoted H. dim(Λ_{rad}), is the infimum of those δ > 0 such that Λ_{rad} admits coverings ⟨B_{i}⟩ with ∑(diam B_{i})^{δ} → 0.

Combining the results of Sullivan and Bishop-Jones, namely [29, Cor. 4], [32, Thm. 2.17] and [8, Thm 1.1] (generalized to arbitrary d), we may now state:

Theorem 2.1. Any nonelementary complete hyperbolic manifold M = ℍ^{d+1}/Γ satisfies

H. dim(Λ_{rad}) = δ(Γ) = α(Γ)

and

λ_{0}(M) = \begin{cases}d^{2}/4 & if δ(Γ) ≤ d/2 , \\ δ(Γ)(d − δ(Γ)) & if δ(Γ) ≥ d/2 . \end{cases}

For later use we record:

Corollary 2.2. If Γ has a cusp of rank r, then δ(Γ) > r/2.

Proof. Let c ∈ S_{∞}^{d} be a cusp point whose stabilizer contains a subgroup L ≅ ℤ^{r}. Change coordinates so c = ∞ in S_{∞}^{d} = ℝ_{∞}^{d} ∪ {∞}, and so L acts by translations on ℝ^{r} ⊂ ℝ_{∞}^{d}. (In general L can rotate the planes ℝ^{d−r} orthogonal to ℝ^{r}.) Then g ↦ g(0) embeds L as a discrete lattice in ℝ^{r}.

Choose a ball B = B(0, s) ⊂ ℝ_{∞}^{d} with μ(B) > 0, where μ is an invariant density of dimension δ = δ(Γ). Then we have

∑_{g∈L} μ(gB) ≍ ∑ |g′(0)|_{σ}^{δ} < ∞,

where |g′|_{σ} is measured in the spherical metric σ = 2 |dx|/(1 + |x|^{2}). Since

∑_{L} |g′(0)|_{σ}^{δ} = ∑_{L}(1 + |g(0)|^{2})^{−δ} ≍ ∫_{ℝ^{r}}(1 + |x|^{2})^{−δ} dx,

the above integral converges, and thus δ > r/2. □

Here is another useful criterion for the critical exponent [29, Cor. 20]:

Theorem 2.3. If μ is a Γ-invariant density of dimension α, and μ gives positive measure to the radial limit set, then α = δ(Γ).

We also note:

Theorem 2.4. If the Poincaré series diverges at the critical exponent, then any invariant density μ of dimension δ(Γ) is supported on the limit set.

Proof. Let F ⊂ Ω(Γ) be a fundamental domain for the action of Γ. Then

∑_{Γ} μ(gF) = ∑_{Γ} ∫_{F} |g′(x)|_{σ}^{δ} dμ(x) = ∫_{F} P_{δ}(x) dμ(x) ≤ μ(S_{∞}^{d}) < ∞;

since P_{δ}(Γ, x) = ∞ for x ∈ Ω(Γ), we have μ(F) = 0. □

3. Geometrically finite groups

To obtain more precise results, it is useful to impose geometric conditions on the hyperbolic manifold M = ℍ^{d+1}/Γ.

The convex core K(M) is the quotient by Γ of the smallest convex set in ℍ^{d+1} containing all geodesics with both endpoints in the limit set. The manifold M and the group Γ are said to be geometrically finite if the convex core meets the Margulis thick part of M in a compact set. For such a manifold, Λ − Λ_{rad} is equal to the countable set of cusp points (fixed-points of parabolic elements of Γ). If Γ is geometrically finite without cusps, then K(M) is compact, Λ = Λ_{rad} and we say Γ is convex cocompact.

By [31] and Theorem 2.4 we have:

Theorem 3.1. If M = ℍ^{d+1}/Γ is geometrically finite, then

H. dim(Λ_{rad}) = H. dim(Λ).

Moreover, the sphere carries a unique Γ-invariant density μ of dimension δ(Γ) and total mass one; μ is nonatomic and supported on Λ; and the Poincaré series diverges at the critical exponent.

Corollary 3.2. If the limit set of a geometrically finite group is not the whole sphere, then H. dim(Λ) < d.

Proof. Otherwise Lebesgue measure on the sphere would be a second invariant density of dimension δ(Γ) = d. □

Corollary 3.3. Any normalized Γ-invariant density supported on the limit set of a geometrically finite group Γ is either:

• the canonical density of dimension δ(Γ), or

• an atomic measure supported on the cusp points in Λ and of dimension α > δ(Γ).

Proof. If the dimension α of μ is more than δ(Γ), then μ(Λ_{rad}) = 0 by Theorem 2.3, so μ is supported on the countable set of cusps. □

If Γ has cusps then these atomic measures actually exist for any α > δ(Γ) [32, Thm 2.19].

Corollary 3.4. The limit set of a convex cocompact group supports a unique normalized Γ-invariant density.

Recall M = ℍ^{d+1}/Γ is topologically tame if it is homeomorphic to the interior of a compact (d + 1)-manifold.

Theorem 3.5. If M = ℍ^{3}/Γ is geometrically infinite but topologically tame, then

H. dim(Λ_{rad}) = H. dim(Λ) = 2.

Proof. In [13] it is shown that λ_{0}(M) = 0 when M is geometrically infinite but topologically tame. Since λ_{0} = δ(2−δ) and δ > 0, the result follows from Theorem 2.1. □

Non-uniqueness of μ for topologically tame manifolds. When M^{3} = ℍ^{3}/Γ is topologically tame but geometrically infinite, there may be more than one normalized invariant density μ in the critical dimension δ(Γ) = 2.

To sketch an example, we start with any compact, acylindrical, atoroidal 3-manifold N such that ∂N has 3 components. By Thurston’s hyperbolization theorem, N admits convex hyperbolic structures M(X, Y, Z) parameterized by conformal structures (X, Y, Z) on the pieces of ∂N. (See, e.g. [26, §5].)

Let ϕ and ψ be pseudo-Anosov mapping classes, and let M be any algebraic limit of M_{n} = M(ϕ^{n}(X), ψ^{n}(Y), Z) (such limits exist by compactness of AH(N) [35]). Then M = ℍ^{3}/Γ has two geometrically infinite and asymptotically period ends E_{1} and E_{2}, as well as a geometrically finite end corresponding to Z. (Compare [23, §3].)

Let g_{n} : M → (0, ∞] be the Green’s with a pole at p_{n} → ∞ in E_{1}, scaled so g_{n}(∗) = 1 at a fixed basepoint. By Harnack’s principle there is a convergent subsequence, with limit a positive harmonic function h_{1}. The function h_{1} tends to infinity in the end E_{1}, is bounded in E_{2} and tends to zero at Z (compare [30]). There is a similar positive harmonic function h_{2} tending to infinity in E_{2} and bounded in E_{1}. Then h_{1} and h_{2} are linearly independent, and at infinity they determine mutually singular invariant densities μ_{1} and μ_{2} of dimension two.

It seems likely that for topologically tame hyperbolic 3-manifolds, the space of invariant densities is finite-dimensional and its dimension is controlled by the number of ends of M.

Notes and references.

1. The unique density guaranteed in Theorem 3.1 can be related to the Hausdorff measure or packing measure of the limit set in dimension δ(Γ) [31].

2. Some papers we cite define geometric finiteness in terms of a convex fundamental polyhedron. This definition agrees with ours when dim(M) ≤ 3, and the results we quote remain valid with the present definition. See [10] for a discussion of the definition of geometric finiteness.

3. It is conjectured that M = ℍ^{3}/Γ is topologically tame whenever Γ is finitely generated, and tameness is known in many cases [9], [14]. So Theorem 3.5 suggests:

Conjecture 3.6. For any finitely generated Kleinian group Γ ⊂ Isom(ℍ^{3}), we have H. dim(Λ_{rad}) = H. dim(Λ).

Any counterexample to this conjecture must have a limit set of positive area [8, Thm 1.7].

1. See also [5], [27], and [38] for results treated in this section.

4. Cores and geometric limits

In this section we introduce the algebraic, geometric and strong topologies on the space of all Kleinian groups. The main result is the following criterion for the limit set and the truncated convex core to move continuously.

Theorem 4.1 (Convergence of cores). Suppose Γ_{n} → Γ strongly, where Γ is geometrically finite. Then:

1. The manifold M_{n} = ℍ^{d+1}/Γ_{n} is geometrically finite for all n ≫ 0, 2. The limit sets satisfy Λ_{n} → Λ in the Hausdorff topology, and

3. The truncated convex cores of the quotient manifolds satisfy K_{ϵ}(M_{n}) → K_{ϵ}(M) strongly, for all ϵ > 0.

Here is a criterion to promote algebraic convergence to strong convergence:

Theorem 4.2. Let Γ_{n} → Γ_{A} algebraically, where Γ_{A} is geometrically finite. Then Γ_{n} → Γ_{A} strongly if and only if L_{n} → L_{A} geometrically for each maximal parabolic subgroup L_{A} ⊂ Γ_{A}.

In the statement above, L_{n} = χ_{n}(L_{A}) are the subgroups of Γ_{n} corresponding algebraically to L_{A}.

Corollary 4.3. If Γ_{A} is convex cocompact, then algebraic convergence implies strong convergence.

Algebraic and geometric limits. Let Γ_{n} ⊂ Isom(ℍ^{d+1}) be a sequence of Kleinian groups. There are several possible notions of convergence of the sequence Γ_{n}. We say Γ_{n} → Γ_{G} geometrically if we have convergence in the Hausdorff topology on closed subsets of Isom(ℍ^{d+1}). We say Γ_{n} → Γ_{A} algebraically if there exist isomorphisms χ_{n} : Γ_{A} → Γ_{n} such that χ_{n}(g) → g for each g ∈ Γ_{A}.

A sequence Γ_{n} has at most one geometric limit Γ_{G}, but it may have many algebraic limits Γ_{A} (coming from different ‘markings’ of Γ_{n}). If the geometric limit exists, then it contains all the algebraic limits.

Here is a description of geometric convergence from the point of view of quotient manifolds. By choosing a standard baseframe at one point in ℍ^{d+1}, we obtain a bijective correspondence between (torsion-free) Kleinian groups and baseframed hyperbolic manifolds (M, ω). Then

(M_{n}, ω_{n}) → (M, ω)

geometrically if and only if, for each compact submanifold K ⊂ M containing ω, there are smooth embeddings ϕ_{n} : K → M_{n} for n ≫ 0 such that ϕ_{n} sends ω to ω_{n} and ϕ_{n} tends to an isometry in the C^{∞} topology. (See [6, Thm. E.1.13].)

Strong convergence. We say Γ_{n} → Γ_{S} strongly if

(a) Γ_{n} → Γ_{S} geometrically, and

(b) for n ≫ 0 there exist surjective homomorphisms

χ_{n} : Γ_{S} → Γ_{n}

such that χ_{n}(g) → g for all g ∈ Γ_{S}.

If Γ_{n} converges to Γ both geometrically and algebraically, then it converges strongly. However strong convergence is more general, since we require only a surjection (instead of an isomorphism) in (b). This generality accommodates situations like Dehn filling in 3-manifolds.

Note that when Γ_{S} is finitely generated, any two choices for χ_{n} in (b) agree for all n ≫ 0.

Lemma 4.4. Suppose Γ_{n} has algebraic and geometric limits Γ_{A} ⊂ Γ_{G}, and M_{G} = ℍ^{d+1}/Γ_{G} is topologically tame. Then Γ_{n} → Γ_{G} strongly.

Proof. By tameness there is a compact submanifold K ⊂ M_{G} such that the inclusion induces an isomorphism on π_{1}. Then we may identify π_{1}(K) with Γ_{G}. Geometric convergence provides nearly isometric embeddings ϕ_{n} : K → M_{n} = ℍ^{d+1}/Γ_{n} for n ≫ 0; on the level of π_{1}, these give homomorphisms χ_{n} : Γ_{G} → Γ_{n} converging to the identity.

By algebraic convergence, there are also isomorphisms χ′_{n} : Γ_{A} → Γ_{n} converging to the identity. Since Γ_{A} ⊂ Γ_{G} and χ_{n}|Γ_{A} = χ′_{n} for all n ≫ 0, the maps χ_{n} are surjective. □

Lemma 4.5. Let Γ_{n} → Γ_{A} algebraically. Then Γ_{n} → Γ_{A} strongly iff for all sequences g_{n} in Γ_{A},

(4.1) (g_{n} → ∞ in Γ_{A}) ⟹ (χ_{n}(g_{n}) → ∞ in Isom(ℍ^{d+1}))

where χ_{n} : Γ_{A} → Γ_{n} are isomorphisms converging to the identity.

Remark. We say x_{n} → ∞ in X if {x_{n}} ∩ K is finite for every compact K ⊂ X.

Proof. Assume (4.1), and consider any h ∈ Isom(ℍ^{d+1}) on which Γ_{n} accumulates. Then passing to a subsequence we can write h = lim χ_{n}(g_{n}) ∈ Γ_{n}. By (4.1) g_{n} returns infinitely often to a compact (hence finite) subset of Γ_{A}, so g_{n} equals some fixed g for infinitely many n. Then h = lim χ_{n}(g) = g, and therefore Γ_{G} = Γ_{A}.

Conversely, suppose Γ_{n} converges geometrically to Γ_{A}. Consider any g_{n} ∈ Γ_{A} such that χ_{n}(g_{n}) ↛ ∞. Then χ_{n}(g_{n}) → h along a subsequence, so h ∈ Γ_{A} and χ_{n}(h^{−1}g_{n}) → id. Therefore g_{n} = h for all n ≫ 0 and g_{n} ↛ ∞, as required by (4.1). □

The truncated convex core. We now turn to an analysis of convergence when the limiting manifold is geometrically finite.

Given a hyperbolic manifold M, let K(M) denote its convex core, and for ϵ > 0 let M^{<ϵ} denote the ϵ-thin part of M (where the injectivity radius is less than ϵ). The truncated core is defined by

K_{ϵ}(M) = K(M) − M^{<ϵ}.

Note that M is geometrically finite iff K_{ϵ}(M) is compact when ϵ is the Margulis constant. If M is geometrically finite, and ϵ is less than both the Margulis constant and the length of the shortest geodesic on M, then there is a retraction

ρ : M → K_{ϵ}(M).

(First take the nearest point projection to K(M), and then use the product structure in the cusps.)

Proof of Theorem 4.2. Clearly strong convergence of Γ_{n} to Γ_{A} implies strong convergence of parabolic subgroups.

Now assume L_{n} → L_{A} for each maximal parabolic subgroup L_{A} ⊂ Γ_{A}. We will show Γ_{n} → Γ_{A} geometrically, and hence strongly.

Pass to any subsequence such that the geometric limit Γ_{G} of Γ_{n} exists. Then Γ_{A} is a subgroup of Γ_{G}, so on the level of quotient manifolds we have the diagram:

\begin{matrix}M_{A} \\ π & ↓ \\ M_{G} & −−→^{ϕ_{n}} & M_{n} \end{matrix}

where π is a covering map and ϕ_{n} are nearly isometric embeddings defined on larger and larger compact submanifolds of M_{G} (as suggested by the notation ϕ_{n} : M_{G} −−→ M_{n}). Each manifold is equipped with a baseframe, chosen for simplicity in its convex core, and the maps above preserve baseframes.

The convex cocompact case. To give the idea of the proof, first consider the case where Γ_{A} is convex cocompact. Then K(M_{A}) is compact and homotopy equivalent to M_{A}. The composition ϕ_{n} ∘ π is C^{∞} close to a (local) isometry on K(M_{A}) for all n ≫ 0, and the isomorphisms

χ_{n} : Γ_{A} → Γ_{n}

converging to the identity are the same as the maps on fundamental group

(ϕ_{n} ∘ π)_{∗} : π_{1}(K(M_{A}), ∗) → π_{1}(M_{n}, ∗)

defined for all n ≫ 0.

Let g_{n} → ∞ in Γ_{A}, and let γ_{n} ⊂ K(M_{A}) be the geodesic paths beginning and ending at the basepoint and representing g_{n} ∈ π_{1}(M_{A}, ∗). Then γ′_{n} = ϕ_{n} ∘ π(γ_{n}) has small geodesic curvature and length comparable to γ_{n}, so the geodesic representative of γ′_{n} is also long. Therefore χ_{n}(g_{n}) → ∞ in Isom(ℍ^{d+1}). By Lemma 4.5 above, Γ_{n} → Γ_{A} geometrically, and hence strongly.

The geometrically finite case. Now suppose only that Γ_{A} is geometrically finite. Choose ϵ > 0 less than both the Margulis constant and the length of the shortest geodesic on M_{A}. As noted above, there is a smooth retraction

ρ : M_{A} → K_{ϵ}(M_{A})

from the manifold to its truncated core.

Consider again the geodesic representatives γ_{n} ⊂ M_{A} of a sequence g_{n} → ∞ in Γ_{A}. Focusing attention on a particular n ≫ 0, write

γ_{n} = δ ∪ ξ_{1} ∪ ⋯ ∪ ξ_{s}, (4.2)

where δ = γ ∩ K_{ϵ}(M_{A}), and the {ξ_{i}} are geodesic segments in M_{A}^{≤ϵ}. These segments account for excursions of γ_{n} into the cusps of M_{A}. Modify this decomposition slightly by absorbing into δ any short ξ_{i} (say of length less than 1).

Let δ′ = ϕ_{n} ∘ π(δ), and define ξ′_{i} ⊂ M_{n} by first retracting ξ_{i} to the truncated core K_{ϵ}(M_{A}), then straightening ϕ_{n}∘π∘ρ(ξ_{i}) rel its endpoints. Then

γ′_{n} = δ′ ∪ ξ′_{1} ∪ ⋯ ∪ ξ′_{s}

is a based piecewise geodesic segment representing the homotopy class χ_{n}(g_{n}) in M_{n}. See Figure 1.

Strong convergence of cusps (the condition L_{n} → L_{A} geometrically) implies the length of ξ′_{i} tends to infinity as the length of ξ_{i} tends to infinity, by Lemma 4.5.

We claim γ′_{n} is nearly a geodesic. More precisely, the geodesic segments making up γ′_{n} have definite length and meet in small angles, and

ϕ_{n} ∘ π  γ_{n} M_{A} K_{ϵ}(M_{A}) M_{n}  γ′_{n}

Figure 1. A piecewise-geodesic representative of γ_{n} in M_{n}.

these angles tend to zero as n → ∞. Indeed, if one such segment ξ_{i} is very long, then ξ′_{i} is also very long, so ξ′_{i} and δ′ are both nearly perpendicular to the boundary of the thin part, hence nearly parallel. On the other hand, if ξ_{i} has only moderate length, then ξ_{i} is within a compact neighborhood of K_{ϵ}(M_{A}). Hence ξ′_{i} ∪δ′ is close to ϕ_{n} ∘π(ξ_{i} ∪δ), so there is a small angle in this case too.

Now as n → ∞, we have ℓ(γ_{n}) → ∞, so in the decomposition (4.2) we either have a large number of segments or a long individual segment. In either case the same is true of γ′_{n}, so ℓ(γ′_{n}) → ∞. Therefore χ_{n}(g_{n}) → ∞ and we have again established strong convergence by Lemma 4.5.

□

Convergence of cores. Next we explain the condition on cores in the statement of Theorem 4.1. Suppose (M_{n}, ω_{n}) → (M, ω) geometrically, and ϕ_{n} : M −−→ M_{n} are almost-isometries defined on larger and larger compact submanifolds. If K_{n} ⊂ M_{n}, K ⊂ M are compact sets, we say K_{n} → K strongly if:

(i) K_{n} is contained in a unit neighborhood of ϕ_{n}(K) for all n ≫ 0, and

(ii) ϕ_{n}^{−1}(K_{n}) → K in the Hausdorff topology on compact subsets of M.

Note that (i) prevents any part of K_{n} from disappearing in the limit by tending to infinity. An equivalent formulation is that the Hausdorff distance between K_{n} and ϕ_{n}(K) tends to zero.

For later reference we quote [23, Prop. 2.4]:

Proposition 4.6. If Γ_{n} → Γ geometrically, and the injectivity radius of the quotient manifold M_{n} in its convex core K(M_{n}) is bounded above, independent of n, then

Λ(Γ_{n}) → Λ(Γ)

in the Hausdorff topology.

Proof of Theorem 4.1 (Convergence of cores). We first show K_{ϵ}(M_{n}) → K_{ϵ}(M) strongly. The argument is very similar to the proof of Theorem 4.2.

Consider any ϵ > 0 less than the Margulis constant and the length of the shortest geodesic on M. As before there is a retraction

ρ : M → K_{ϵ}(M),

and we can identify Γ with π_{1}(K_{ϵ}(M)). By strong convergence, the nearly isometric embedding

ϕ_{n} : K_{ϵ}(M) → M_{n}

determines a surjective homomorphism

χ_{n} : Γ → Γ_{n}

for all n ≫ 0. Moreover, ϕ_{n} sends the thin part into the thin part, so geodesics passing through the thick part of M_{n} are represented by geodesics on M.

Consider any closed geodesic γ_{n} passing through the ϵ-thick part of M_{n}. Let γ be the shortest closed geodesic in M such that ϕ_{n}(ρ(γ)) is homotopic to γ_{n}. As before, we can write

γ = δ ∪ ξ_{1} ∪ ⋯ ∪ ξ_{s},

where δ = γ ∩K_{ϵ}(M), and the {ξ_{i}} are geodesic segments in M^{≤ϵ}. Then γ_{n} is homotopic to the broken geodesic

γ′_{n} = δ′ ∪ ξ′_{1} ∪ ⋯ ∪ ξ′_{s},

where δ′ = ϕ_{n}(δ) and ξ′_{i} is obtained by straightening ϕ_{n} ∘ ρ(ξ_{i}) while holding its endpoints fixed.

Now we claim ξ′_{i} is long whenever ξ_{i} is long. Indeed, for any R > 0 there exists an N(R) such that ϕ_{n} is defined and nearly isometric on an R-neighborhood of K_{ϵ}(M) when n > N(R). For such n, the straightened segment ξ′_{i} is longer than R/2 whenever ξ_{i} is longer than R; otherwise we could replace ξ_{i} with ϕ_{n}^{−1}(ξ′_{i}) and shorten our representative γ while keeping ϕ_{n}(γ) homotopic to γ_{n}.

Thus one can repeat the proof of Theorem 4.2 to conclude that γ′_{n} is nearly a geodesic, and therefore:

(∗) The loops γ′_{n} and γ_{n} are C^{1}-close in the thick part M_{n}^{≥ϵ}, with a bound depending only on n and tending to zero as n → ∞.

Now any x ∈ K_{ϵ}(M_{n}) lies in the image of an ideal simplex S ⊂ ℍ^{d+1} = M̃_{n} with vertices in the limit set. Approximating the edges of S by lifts of long closed geodesics and applying (∗), we conclude that x lies close to ϕ_{n}(K_{ϵ}(M)).

In particular K_{ϵ}(M_{n}) is compact, so M_{n} is geometrically finite. The same reasoning shows any x ∈ K_{ϵ}(M) is close to ϕ_{n}^{−1}(K_{ϵ}(M_{n})), so the truncated cores converge strongly.

By convergence of cores, the injectivity radius in K(M_{n}) is uniformly bounded above for all n ≫ 0. Thus Proposition 4.6 shows Λ_{n} → Λ. □

Another proof of Theorem 4.1(a), similar in spirit and with many details, can be found in [33].

5. Accidental parabolics

Cusps play a central role in the theory of deformations of geometrically finite manifolds. In this section we discuss the ways in which a closed geodesic can shrink to form a cusp in the algebraic limit of a sequence of hyperbolic 3-manifolds. This shrinking governs the difference between algebraic and strong convergence, and we will prove:

Theorem 5.1. Let Γ_{n} ⊂ Isom(ℍ^{3}) converge algebraically to a geometrically finite group Γ_{A}. Then Γ_{n} → Γ_{A} strongly iff all accidental parabolics converge horocyclically.

On the other hand, we will see in §7 and §8 that the stronger condition of radial convergence is required to obtain convergence of the dimension of the limit set.

1  1

Figure 2. Radial and horocyclic convergence of multipliers.

Radial and horocyclically convergence. Consider a sequence t_{n} = iL_{n} + θ_{n} in the upper half-plane ℍ ⊂ ℂ with t_{n} → 0. We say t_{n} → 0 radially if there exists an M such that

|θ_{n}|/L_{n} < M

for all n, while t_{n} → 0 horocyclically if

θ_{n}^{2}/L_{n} → 0.

Radial convergence means t_{n} remains within a bounded hyperbolic distance of a geodesic converging to t = 0, while horocyclic convergence means any horoball resting on t = 0 contains t_{n} for all n ≫ 0.

Now suppose λ_{n} → 1 in ℂ^{∗} and |λ_{n}| > 1. We say λ_{n} → 1 radially or horocyclically if t_{n} = i log λ_{n} → 0 radially or horocyclically in ℍ. See Figure 2. Radial convergence is equivalent to the condition

|λ_{n} − 1| ≤ M||λ_{n}| − 1|

for some M.

The complex length of a hyperbolic element g ∈ Isom^{+}(ℍ^{3}) is given by

L(g) = L + iθ = log λ,

where the multiplier λ = g′(x) is the derivative of g at its repelling fixed-point. The real part L is the length of the core geodesic of the solid torus ℍ^{3}/⟨g⟩, and θ is the torsion of parallel transport around this geodesic. When g is nearly parabolic (λ is close to 1) there is a natural choice of θ close to 0.

Now suppose Γ_{n} → Γ_{A} algebraically, with isomorphisms χ_{n} : Γ_{A} → Γ_{n} converging to the identity. An accidental parabolic g ∈ Γ_{A} is a parabolic element such that g_{n} = χ_{n}(g) is hyperbolic for infinitely many n. The complex lengths of these g_{n} satisfy

iL(g_{n}) → 0 in ℍ.

If the convergence above is radial (or horocyclic), we say all accidental parabolics converge radially (or horocyclically). (For Γ_{A} ⊄ Isom^{+}(ℍ^{3}) we apply the condition to its orientation preserving subgroup.)

Proof of Theorem 5.1. Suppose Γ_{n} → Γ_{A} algebraically. By Lemma 4.5, Γ_{n} → Γ_{A} strongly iff L_{n} → L_{A} geometrically for each maximal parabolic subgroup L_{A} ⊂ Γ_{A}, where L_{n} = χ_{n}(L_{A}).

If L_{n} is a parabolic subgroup for all n ≫ 0, then it is conjugate to a group of translations on ℂ, and geometric convergence is easily verified (cf. Theorem 6.2 below).

Otherwise L_{n} is hyperbolic for infinitely many n, so L_{A} is of rank-1 and generated by an accidental parabolic. Pass to the subsequence where L_{n} is hyperbolic, and let λ_{n} → 1 denote the multiplier of a generator of L_{n}. To analyze the geometric limit in this case, consider the quotient torus

X_{n} = Ω(L_{n})/L_{n} ≅ ℂ^{∗}/λ_{n}^{ℤ} ≅ ℂ/(ℤ2πi ⊕ ℤ log λ)

and the quotient cylinder

X_{A} = Ω(L_{A})/L_{A} ≅ ℂ/ℤ.

Suppose λ_{n} → 1 horocyclically. Then t_{n} = i(log λ)/2π → 0 horocyclically in ℍ. But this means [X_{n}] → ∞ in the moduli space of complex tori M_{1} = ℍ/SL_{2}(ℤ), since horoballs resting on z = 0 form neighborhoods of the cusp of the modular surface. Thus X_{n} becomes long and thin as n → ∞, so it converges geometrically (as a surface with a complex affine structure) to the cylinder X_{A}. Therefore L_{n} → L_{A} geometrically as well.

On the other hand, if the convergence is not horocyclic, then after passing to a subsequence the tori [X_{n}] converge to a torus X_{G} ∈ M_{1}. Thus L_{n} converges geometrically to a rank-2 parabolic subgroup L_{G}, with

X_{G} ≅ Ω(L_{G})/L_{G}.

Hence L_{G} ≠ L_{A} and strong convergence fails whenever horocyclic convergence fails. □

6. Cusps and Poincaré series

In this section we continue our study of cusps from a more analytical point of view. We consider a single parabolic group L and a deformation given by a family of representations χ_{n} : L → L_{n} converging to the identity. The results we develop control the Poincaré series for L_{n} as n → ∞.

This control will be applied in the next section to establish continuity of the Hausdorff dimension of the limit set.

Deformations of cusps. Let L be an elementary Kleinian group. Then L is a finite extension of a free abelian group ℤ^{r}, and we set rank(L) = r. We say L is hyperbolic if it contains a hyperbolic element; in this case rank(L) = 1. Otherwise L is parabolic, its elements share a common fixed-point c, |g′(c)| = 1 for all g ∈ L and 0 ≤ rank(L) ≤ d.

Now consider a deformation given by a sequence of representations χ_{n} : L → L_{n}, and a sequence of real numbers δ_{n}, satisfying the following conditions:

1. L_{n} and L are elementary Kleinian groups fixing a common point c ∈ S_{∞}^{d};

2. L is parabolic, with rank(L) ≥ 1;

3. χ_{n} : L → L_{n} is a surjective homomorphism, converging pointwise to the identity; and

4. δ_{n} → δ > rank(L)/2.

Example: Dehn filling. As a typical example, these conditions arise naturally when one performs (p_{n}, q_{n}) Dehn-filling on a rank two parabolic subgroup L of a Kleinian group Γ [34, Ch. 4], [6, E.5-E.6]. In the filled group Γ_{n}, the cusp becomes a short geodesic with stabilizer L_{n} = ⟨g_{n}⟩, and there is a surjective filling homomorphism χ_{n} : L → L_{n}. This example shows we can have rank(L_{n}) < rank(L) for all n. In addition, L_{n} need not have any algebraic limit; if p_{n} and q_{n} tend to infinity, then g_{n} → ∞ in Isom(ℍ^{3}) (although L_{n} converges to L geometrically). In our applications the exponents will be given by δ_{n} = δ(Γ_{n}).

Uniform convergence of Poincaré series. Recall that for Γ ⊂ Isom(ℍ^{d+1}), x ∈ S_{∞}^{d} and δ ≥ 0, the absolute Poincaré series is defined by

P_{δ}(Γ, x) = ∑_{g∈Γ} |g′(x)|_{σ}^{δ},

where the derivative is measured in the spherical metric σ. To study the rate of convergence we define for any open set U the sub-sum

P_{δ}(Γ, U, x) = ∑_{g(x)∈U} |g′(x)|_{σ}^{δ}.

Let us say the Poincaré series for (L_{n}, δ_{n}) converge uniformly if for any compact set K ⊂ S_{∞}^{d} − {c} and ϵ > 0, there is a neighborhood U of c such that

P_{δ_{n}}(L_{n}, U, x) < ϵ

for all n ≫ 0 and all x ∈ K. This means the tail of the series can be made small, independent of n, by choosing U small enough.

We will establish uniform convergence under the following 3 conditions.

Theorem 6.1. If L_{n} → L geometrically and

δ > \begin{cases}1 & if d = 2, or \\ (d − 1)/2 & if d ≠ 2, \end{cases}

then the Poincaré series for (L_{n}, δ_{n}) converge uniformly.

Theorem 6.2. If L_{n} is parabolic with rank(L_{n}) ≥ d − 1 for all n, then L_{n} → L strongly and the Poincaré series converge uniformly.

Theorem 6.3. If L_{n} → L ⊂ Isom(ℍ^{3}) algebraically, L_{n} are hyperbolic and all accidental parabolics converge radially, then L_{n} → L strongly and the Poincaré series converge uniformly.

Cusps in high-dimensional manifolds. We remark that the conclusion of Theorem 6.2 holds for all cusps in the case of hyperbolic 3-manifolds (since d = 2), but it fails for cusps of low rank in highdimensional hyperbolic manifolds (d > 2). A related fact is that the rank of a cusp can increase in the geometric limit.

For a standard example, let R_{n} be the isometry of ℝ_{∞}^{3} given by rotating angle 2π/n about the line (x = 0, y = n); let T_{n}(x, y, z) = (x, y, z+1/n); and let g_{n} = T_{n}∘R_{n}. Then L_{n} = ⟨g_{n}⟩ is a rank 1 parabolic group, converging geometrically to the rank 2 cusp L_{G} = ⟨g, h⟩ where

g(x, y, z) = lim g_{n}(x, y, z) = (x + 2π, y, z),
h(x, y, z) = lim g_{n}^{n}(x, y, z) = (x, y, z + 1).

The algebraic limit L = ⟨g⟩ also exists and with the obvious isomorphism χ_{n} : L → L_{n} and δ_{n} = δ = 1/2+ϵ we have the setup for Theorem 6.2, except rank(L_{n}) = d − 2. But L_{n} does not converge strongly to L (because L ≠ L_{G}), and the Poincaré series do not converge uniformly (because P_{δ}(L_{G}, x) = ∞).

Proof of Theorem 6.1. For simplicity, assume K = {p} is a single point; the case of a general compact set is similar.

Normalize coordinates on S_{∞}^{d} = ℝ_{∞}^{d} ∪ {∞} so c = 0 and Lp ⊂ ℝ_{∞}^{d}. Since L is parabolic, it acts freely and properly discontinuously on S_{∞}^{d} − {c}. Thus we can choose a ball B = B(p, r), r ≪ |p|, such that its translates L · B are bounded and disjoint. Since L_{n} → L geometrically, we can also arrange that the balls L_{n} ·B are bounded and disjoint for all n ≫ 0.

To give the main idea of the proof, we first treat the case where we have δ > d/2 and L_{n} is parabolic for all n. Then all L_{n} act isometrically with respect to the metric

ρ = \frac{|dx|}{|x|^{2}}

obtained by pulling back the Euclidean metric |dx| under an inversion sending the cusp point c to ∞. Consequently diam(gB) ≍ d(0, gB)^{2}. Thus

P_{δ}(L, p) = ∑_{L} |g′(p)|_{σ}^{δ} ≍ ∑ diam(gB)^{δ} = ∑ diam(gB)^{d} diam(gB)^{δ−d}
≍ ∑ ∫_{gB} |x|^{2(δ−d)} |dx|^{d} = ∫_{⋃ gB} |x|^{2(δ−d)} |dx|^{d} < ∞,

because

2(δ − d) > 2(d/2 − d) = −d,

and −d is the critical exponent for integrability of the singularity |x|^{α} on ℝ^{d}.

To obtain uniform convergence, choose α such that

2(δ_{n} − d) > α > −d

for all n ≫ 0. Consider a small neighborhood U = B(0, s) of the cusp point c. Then for all n ≫ 0 we have similarly

P_{δ_{n}}(L_{n}, U, p) ≍ ∫_{⋃{gB : gp∈U}} |x|^{2(δ_{n}−d)} |dx|^{d}
= O (∫_{U} |x|^{α} |dx|^{d}) = O(s^{d+α}).

Since d + α > 0, this bound tends to zero as s → 0 and we have established uniform convergence of the Poincaré series for (L_{n}, δ_{n}).

Next we treat the case where L_{n} is hyperbolic. Then the geodesic stabilized by L_{n} has endpoints {a_{n}, c} on the sphere at infinity. In this case the invariant metric becomes

ρ_{n} = \frac{|dx|}{|x − a_{n}||x|},

and we have the estimate

diam(B) ≍ d(0, B)d(a_{n}, B).

The calculation above becomes

P_{δ_{n}}(L_{n}, U, p) = O (∫_{U} |x − a_{n}|^{α/2}|x|^{α/2} |dx|^{d})
= O (∫_{U} |x − a_{n}|^{α} + |x|^{α} |dx|^{d}) = O(s^{d+α}),

and as before this bound gives uniform convergence.

Since d/2 = 1 when d = 2, the proof is now complete in the case of classical Kleinian groups. Also when d = 1 we have

δ > rank(L)/2 = 1/2 = d/2,

so we have covered this case as well.

Finally we explain how d/2 can be improved to (d − 1)/2 for d ≥ 3. Replacing L with a suitable abelian subgroup of finite index, we can assume that for each n there is a maximal connected abelian Lie group G_{n} with

L_{n} ⊂ G_{n} ⊂ Isom^{+}(ℍ^{d+1}).

Fixing attention on a particular n, let D = dim(G_{n}). If L_{n} is hyperbolic, then G_{n} is conjugate into ℝ^{∗} × SO(d) and we have

D = 1 + [d/2],

since [d/2] is the dimension of a maximal torus in SO(d). If L_{n} is parabolic of rank r, then G_{n} is conjugate into ℝ^{r} × SO(d − r), so we have

D = r + [(d − r)/2] = [(d + r)/2].

Now for D < d the orbit L_{n}p is rather sparse, since it is confined to the D-dimensional submanifold G_{n}p. Due to this sparseness, the critical dimension for integrability of |x|^{α} becomes −D instead of −d. Thus we need only guarantee δ > D/2 to achieve uniform convergence of the Poincaré series. (For a detailed proof of this statement, it is useful to change coordinates so c = ∞ and p = 0. Moving p distance ≤ 1 if necessary, we can assume the stabilizer of p in G_{n} is trivial, and that the injectivity radius of the submanifold G_{n}p ⊂ ℝ_{∞}^{d} at p is bounded below. Then for L_{n} parabolic the orbit of p in ℝ_{∞}^{d} is a cylinder:

G_{n}p ≅ (S^{1})^{D−r} × ℝ^{r},

while for L_{n} hyperbolic it is a cone with vertex a_{n}:

G_{n}p ≅ (S^{1})^{D−1} × ℝ_{+}.

The bound on the Poincaré series becomes

∫_{G_{n}p} \frac{|dx|^{D}}{(1 + |x|^{2})^{δ}} ,

and this integral is uniformly bounded, independent of the shape of the cylinder or cone, if δ > D/2. To see this uniformity, compare the integral above to one where G_{n}p is replaced by a D-plane through p.)

It remains only to check that δ > (d − 1)/2 implies δ > D/2. If L_{n} is hyperbolic then d ≥ 3 implies

D ≤ 1 + [d/2] ≤ d − 1,

while if L_{n} is parabolic of rank r ≤ d − 1 we have

D ≤ [(2d − 1)/2] = d − 1;

in either case, we have δ > (d − 1)/2 ≥ D/2 as desired. Finally if rank(L_{n}) = d then D = d and we have

δ > rank(L)/2 = d/2 = D/2,

so we are done. □

Proof of Theorem 6.2. As before we assume K = {p} is a single point. Normalize coordinates so c = ∞ and p = 0, and pass to a subgroup of index two if necessary, so L and L_{n} preserve orientation for all n ≫ 0. Then the rank restriction implies L_{n} and L act by pure translations on ℝ_{∞}^{d}. Thus for g ∈ L we can write g(z) = z + ℓ(g) and (χ_{n}g)(z) = z + ℓ_{n}(g).

Since χ_{n} converges to the identity, the length ratio satisfies

sup_{L−\{0\}} \frac{|ℓ_{n}(g)|}{|ℓ(g)|} → 1 (6.1)

as n → ∞. In particular, χ_{n} is an isomorphism for all n ≫ 0, so L_{n} → L algebraically. Moreover large elements of L map to large elements of L_{n}, so L_{n} → L strongly (by Lemma 4.5).

Now recall the spherical metric is given by σ = 2|dx|/(1 + |x|^{2}). Let U = {z : |z| > R} be a neighborhood of ∞. Then by (6.1), for n ≫ 0, the Poincaré series satisfies

P_{δ_{n}}(L_{n}, U, p) = ∑_{g(p)∈U} |g′(p)|_{σ}^{δ_{n}} = ∑_{\{g∈L : |ℓ_{n}(g)|>R\}} (\frac{1}{1 + |ℓ_{n}(g)|^{2}} )^{δ_{n}}

≤ ∑_{|ℓ_{n}(g)|>R} |ℓ_{n}(g)|^{−2δ_{n}} = O ( ∑_{|ℓ(g)|>R/2} |ℓ(g)|^{−2α}) ,

where α is chosen so δ_{n} > α > rank(L)/2 for all n ≫ 0. Since the last sum converges, it can be made arbitrarily small by a suitable choice of R. Thus the Poincaré series converges uniformly. □

Proof of Theorem 6.3. Once again we assume K = {p}. By algebraic convergence, we can write L = ⟨g⟩ and χ_{n}(g) = g_{n}. Normalize coordinates so c = ∞, p = 0, and g(p) = 1, where we have identified ℝ_{∞}^{2} with ℂ. Changing L_{n} by a conjugacy tending to the identity, we can assume

g(z) = z + 1,
g_{n}(z) = λ_{n}z + 1,

and |λ_{n}| > 1.

By hypothesis, |λ_{n} − 1| ≤ M(|λ_{n}| − 1) for some M. So for k > 0, we have

|g_{n}^{k}(0)| = |1 + λ_{n} + λ_{n}^{2} + ⋯ + λ_{n}^{k−1}| = \frac{|λ_{n}^{k} − 1|}{|λ_{n} − 1|}
≥ \frac{|λ_{n}|^{k} − 1}{M(|λ_{n}| − 1)} = \frac{1}{M} (1 + |λ_{n}| + |λ_{n}|^{2} + ⋯ + |λ_{n}|^{k−1})
≥ \frac{k|λ_{n}|^{k/2}}{2M} ·

Since the last bound is independent of n, we see g_{n}^{k} is large whenever k is large, and thus L_{n} → L strongly (by Lemma 4.5). Moreover

∑_{k>K} |(g_{n}^{k})′(0)|_{σ}^{δ_{n}} = ∑_{k>K} (\frac{|λ_{n}|^{k}}{1 + |g_{n}^{k}(0)|^{2}} )^{δ_{n}}

≤ ∑_{k>K} (\frac{(2M)^{2}|λ_{n}|^{k}}{k^{2}|λ_{n}|^{k}} )^{δ_{n}}

= O ( ∑_{k>K} k^{−2α}) ,

where α is chosen so δ_{n} > α > rank(L)/2 = 1/2 for all n ≫ 0. Since the last sum converges, it can be made arbitrarily small by taking K sufficiently large.

Carrying out the same argument with the other fixed-point a_{n} of g_{n} normalized to be at c = ∞, we conclude that for any ϵ > 0 there is a K such that

∑_{|k|>K} |(g_{n}^{k})′(p)|_{σ}^{δ_{n}} < ϵ.

Choose a neighborhood U of c = ∞ such that g_{n}^{k}(p) ∉ U for all n and |k| ≤ K. Then P_{δ_{n}}(L_{n}, U, p) < ϵ, and we have shown that the Poincaré series converge uniformly. □

7. Continuity of Hausdorff dimension

In this section we establish conditions for continuity of the Hausdorff dimension of the limit set.

The continuity of dimension will generally come along with a package of additional properties. For economy of language, we say Γ_{n} → Γ dynamically if:

D1. Γ_{n} → Γ strongly;

D2. The limit sets satisfy Λ_{n} → Λ in the Hausdorff topology on closed subsets of S_{∞}^{d};

D3. H. dim(Λ_{n}) → H. dim(Λ);

D4. The critical exponents satisfy δ(Γ_{n}) → δ(Γ);

D5. The groups Γ_{n} and Γ are geometrically finite for all n ≫ 0; and

D6. The normalized canonical densities satisfy μ_{n} → μ in the weak topology on measures.

Actually when Γ is geometrically finite, conditions D1 and D6 imply the rest (by Theorems 3.1 and 4.1). The terminology is meant to suggest that the dynamical and statistical features of Γ_{n} (as reflected in its limit set and invariant density) converge to those of Γ.

Here is the prototypical example:

Theorem 7.1. If Γ_{n} → Γ algebraically and Γ is convex cocompact, then Γ_{n} → Γ dynamically.

Proof. Since Γ has no parabolic subgroups, Γ_{n} → Γ strongly (Corollary 4.3). By Theorem 4.1, Γ_{n} is geometrically finite for all n ≫ 0 and Λ_{n} → Λ. Now pass to any subsequence such that δ(Γ_{n}) → α and μ_{n} → ν weakly; then ν is a Γ-invariant density supported on the limit set. Such a density is unique for a convex compact group (Corollary 3.4), so ν = μ. We have thus verified D1 and D6, and D2-D5 follow. □

Our goal in this section is to obtain dynamic convergence in the presence of cusps. We will establish the following results.

Theorem 7.2 (Dynamic convergence). Let Γ_{n} ⊂ Isom(ℍ^{d+1}) converge strongly to a geometrically finite group Γ. If

lim inf δ(Γ_{n}) > \begin{cases}1 & if d = 2, or \\ (d − 1)/2 & if d ≠ 2, \end{cases}

then Γ_{n} → Γ dynamically.

For hyperbolic 3-manifolds, there is a simple condition on parabolics that promotes algebraic convergence to dynamic convergence.

Theorem 7.3 (Radial limits). Let Γ_{n} → Γ algebraically, where Γ ⊂ Isom(ℍ^{3}) is geometrically finite. If all accidental parabolics converge radially, then Γ_{n} → Γ dynamically.

Corollary 7.4. If Γ_{n} → Γ is an algebraically convergent sequence of finitely-generated Fuchsian groups, then H. dim(Λ_{n}) → H. dim(Λ).

The bottom of the spectrum λ_{0} is insensitive to subtleties of limit sets of dimension d/2 or less, so from Theorem 7.2 we obtain:

Corollary 7.5. If Γ_{n} converges strongly to a geometrically finite group Γ, then λ_{0}(M_{n}) → λ_{0}(M) for the corresponding quotient manifolds.

Corollary 7.6 (Strong limits). Suppose M = ℍ^{3}/Γ is topologically tame, and Γ_{n} → Γ strongly. Then the quotient manifolds M_{n} = ℍ^{3}/Γ_{n} satisfy

λ_{0}(M_{n}) → λ_{0}(M);

and if H. dim(Λ) ≥ 1, then we also have

H. dim(Λ_{n}) → H. dim(Λ).

In the next section we will give several examples of the discontinuous behavior of dimension. In particular we will show strong convergence alone is insufficient to guarantee H. dim(Λ_{n}) → H. dim(Λ), even for geometrically finite 3-manifolds.

Semicontinuity of dimension. One inequality for the critical dimension and the bottom of the spectrum for a geometric limit is general and immediate:

Theorem 7.7. If Γ_{n} → Γ_{G} geometrically, then

δ(Γ_{G}) ≤ lim inf δ(Γ_{n})  and
λ_{0}(M_{G}) ≥ lim sup λ_{0}(M_{n})

for the corresponding quotient manifolds.

Proof. By Theorem 2.1, the critical dimension δ is the same as the minimal dimension of an invariant conformal density. Thus for each n there exists a normalized Γ_{n}-invariant density μ_{n} of dimension δ(Γ_{n}). Taking the weak limit of a subsequence, we obtain a Γ_{G}-invariant density μ of dimension lim inf δ(Γ_{n}), so δ(Γ_{G}) ≤ lim inf δ(Γ_{n}), again by Theorem 2.1.

Similarly, we can choose an f ∈ C_{0}^{∞}(M_{G}) such that the Ritz-Rayleigh quotient (2.1) is less than λ_{0}(M_{G})+ϵ. This f is supported on a compact submanifold K ⊂ M_{G} that can be nearly isometrically embedded in M_{n} for all n ≫ 0. It follows that lim sup λ_{0}(M_{n}) ≤ λ_{0}(M_{G}) + ϵ; now let ϵ → 0. □

Corollary 7.8. Let M = ℍ^{3}/Γ be a geometrically infinite, topologically tame hyperbolic 3-manifold. Then

δ(Γ_{n}) → δ(Γ) = 2 and
λ_{0}(M_{n}) → λ_{0}(M) = 0

whenever Γ_{n} → Γ algebraically or geometrically.

Proof. We have δ(Γ) = 2 by Theorem 3.5. By the preceding result we have

2 ≥ lim sup δ(Γ_{n}) ≥ lim inf δ(Γ_{n}) ≥ δ(Γ) = 2

if the convergence is geometric. For the case of algebraic convergence, use the fact that any geometric limit Γ′ contains the algebraic limit and thus lim inf δ(Γ_{n}) ≥ δ(Γ′) = δ(Γ) = 2. The relation λ_{0} = δ(2 − δ) completes the proof. □

Proof of Theorem 7.2 (Dynamic convergence). By strong convergence, for n ≫ 0 there are surjective homomorphisms

χ_{n} : Γ → Γ_{n}

converging to the identity. By Theorem 4.1 we also know Γ_{n} is geometrically finite, the limit sets satisfy Λ_{n} → Λ and the truncated cores satisfy K_{ϵ}(M_{n}) → K_{ϵ}(M) strongly.

To complete the proof, we must show (D6): that the canonical measures μ_{n} for Γ_{n} converge weakly to the canonical measure μ for Γ. Pass to any subsequence such that μ_{n} has a weak limit ν, and δ_{n} = δ(Γ_{n}) converges to some limit δ. Clearly ν is a Γ-invariant density of dimension δ.

To prove μ = ν, recall that μ is the unique Γ-invariant density supported on Λ with no atoms at the cusps (Corollary 3.3). Since Λ_{n} → Λ, ν is supported on Λ, so we need only check that ν(c) = 0 for each cusp point c ∈ Λ.

To this end, fix ϵ > 0 and a cusp point c ∈ Λ. We will construct a neighborhood U of c such that μ_{n}(U) < ϵ for all n ≫ 0.

Let L ⊂ Γ be the stabilizer of c and let L_{n} = χ_{n}(L) ⊂ Γ_{n}. Adjusting L_{n} by a conjugacy converging to the identity, we can assume c is also fixed by L_{n}. By Corollary 2.2, we have

δ = lim δ(Γ_{n}) ≥ δ(Γ) > rank(L)/2,

so the sequence (L_{n}, δ_{n}) fits into the setup discussed in §6.

We claim there exists a compact set F ⊂ ℝ^{d} such that

Λ ⊂ L · F ∪ {∞} (7.1)

and, for all n ≫ 0,

Λ_{n} ⊂ L_{n} · F ∪ {∞}. (7.2)

Indeed, by Theorem 4.1 there is an ϵ > 0 less than the Margulis constant such that the truncated core K_{ϵ}(M) is the strong limit of K_{ϵ}(M_{n}). Let H ⊂ ℍ^{d+1} be the horosphere such that H/L ⊂ M is the component of ∂M^{<ϵ} corresponding to c. Projection along geodesics rays from c gives a natural bijection

ℝ_{∞}^{d}/L ≅ H/L ⊂ M.

Since every geodesic from c to Λ lies in the convex hull, this bijection sends Λ/L into K_{ϵ}(M). By compactness of K_{ϵ}(M), there is a compact set F ⊂ ℝ^{d} such that

Λ/L ⊂ F/L ⊂ ℝ_{∞}^{d}/L.

Thus we have (7.1), and strong convergence of K_{ϵ}(M_{n}) to K_{ϵ}(M) implies that a slight enlargement of F satisfies (7.2).

By hypothesis, we have δ > 1 (if d = 2) or δ > (d − 1)/2 (if d ≠ 2). So by Theorem 6.1, the Poincaré series for (L_{n}, δ_{n}) converge uniformly. Since F is compact, this means we can choose a neighborhood U of c such that

P_{δ_{n}}(L_{n}, U, x) < ϵ

for all x ∈ F and n ≫ 0.

Since μ_{n}(c) = 0 and L_{n} · F covers the rest of the limit set Λ_{n}, we have

μ_{n}(U) = μ_{n}(U ∩ (L_{n} · F)) ≤ ∑_{L_{n}} μ_{n}(U ∩ gF)

= ∑_{L_{n}} ∫_{{x∈F : g(x)∈U}} |g′(x)|_{σ}^{δ_{n}} dμ_{n}

= ∫_{F} P_{δ_{n}}(L_{n}, U, x) dμ_{n} ≤ ϵμ_{n}(F) ≤ ϵ.

Since ϵ was arbitrary, the weak limit ν has no atom at c. Thus μ = ν and we have established the dynamic convergence. □

Proof of Theorem 7.3 (Radial limits). We first show Γ_{n} → Γ strongly. Let χ_{n} : Γ → Γ_{n} be isomorphisms converging to the identity, and consider any maximal parabolic subgroup L ⊂ Γ. Set L_{n} = χ_{n}(L) ⊂ Γ_{n}. The cusps of a hyperbolic 3-manifold have rank 1 or 2 (= d or d − 1), so if L_{n} is parabolic it converges strongly to L by Theorem 6.2. Otherwise L is generated by an accidental parabolic; but since we are assuming accidental parabolics converge radially, L_{n} → L strongly by Theorem 6.3. Thus all parabolic subgroups converge strongly, so Γ_{n} → Γ strongly by Theorem 4.2.

We therefore have the setup for Theorem 7.2. As before it suffices to show a weak limit ν of the canonical measure ν_{n} has no mass at a cusp point c. But Theorems 6.2 and 6.3 imply that the Poincaré series converges uniformly, so ν(c) = 0 by the same reasoning as above. □

Proof of Corollary 7.4. Finitely generated Fuchsian groups are always geometrically finite, and all accidental parabolics converge radially (since the multipliers are in ℝ^{∗}). Thus Theorem 7.3 applies. □

Proof of Corollary 7.5. Consider any subsequence such that λ_{0}(M_{n}) converges to a limiting value λ. If λ = d^{2}/4, then

\frac{d^{2}}{4} ≥ λ_{0}(M_{G}) ≥ lim λ_{0}(M_{n}) = \frac{d^{2}}{4}

by Theorem 7.7, so we have convergence of λ_{0}. Otherwise λ < d^{2}/4 and the relation λ_{0} = δ(d − δ) (Theorem 2.1) shows

lim δ(Γ_{n}) = λ(d − λ) > d/2.

By Theorem 7.2, δ(Γ_{n}) → δ(Γ_{G}) and thus λ_{0}(M_{n}) → λ(M_{G}) in this case as well. □

Proof of Corollary 7.6 (Strong limits). First suppose M is geometrically finite. Then the convergence of λ_{0} is contained in Corollary 7.5. Since

1 ≤ H. dim(Λ) ≤ lim inf H. dim(Λ_{n}),

and M_{n} is geometrically finite for all n ≫ 0, H. dim(Λ_{n}) is determined by λ_{0}(M_{n}) (via the relation λ_{0} = δ(2 − δ)), so the dimensions of the limit sets also converge.

For M geometrically infinite, the convergence of dimension and λ_{0} follows from Corollary 7.8. □

8. Examples of discontinuity

In this section we sketch three examples of the discontinuous behavior of the δ(Γ) as Γ varies. We only consider geometrically finite groups; thus we have δ(Γ) = H. dim(Λ(Γ)) throughout, and so these examples also demonstrate discontinuity of the dimension of the limit set.

I. δ(Γ) is not continuous in the geometric topology. Let Γ_{0} ⊂ Isom(ℍ^{d+1}) be a cocompact Kleinian group. By a result of Mal’cev, any finitely generated linear group is residually finite [21, Thm. VII]. Thus there is a descending sequence of subgroups of finite index,

Γ_{0} ⊃ Γ_{1} ⊃ Γ_{2} …,

such that ⋂ Γ_{n} = Γ = {1}.

The limit sets of these groups satisfy Λ(Γ_{n}) = S_{∞}^{d} for all n, while Λ(Γ) = ∅. So although we have Γ_{n} → Γ geometrically, the critical dimension δ(Γ_{n}) = d does not converge to δ(Γ) = 0.

Note that the strong convergence fails dramatically in this example, since there is no surjection Γ → Γ_{n}.

II. Convergence of limit sets vs. convergence of δ(Γ). Here is an example where δ is discontinuous even though Λ varies continuously.

Consider a sequence of open hyperbolic Riemann surfaces M_{n} of genus two with one end of infinite area. Let γ_{n} be a simple geodesic separating M_{n} into two subsurfaces X_{n}, Y_{n} of genus one, with area(X_{n}) = ∞ and area(Y_{n}) = π. Suppose that as n → ∞ the length of γ_{n} tends to zero (see Figure 3). Then

λ_{0}(M_{n}) ≤ ∫ |∇ϕ_{n}|^{2}/ ∫ |ϕ_{n}|^{2} → 0,

X_{n}  γ_{n}  Y_{n}

Figure 3. Pinching an open Riemann surface.

where ϕ_{n} is supported on Y_{n}, |∇ϕ_{n}| = 1 on small area neighborhood of γ_{n}, and ϕ_{n} = 1 on the rest of Y_{n}. Thus the limit sets satisfy H. dim(Λ_{n}) → 1.

On the other hand, choosing base-frames in X_{n}, we can arrange the example so (M_{n}, ω_{n}) converges geometrically to a surface (M, ω) of genus one with one cusp and one infinite volume end. The area of K(M_{n}) is constant by Gauss-Bonnet, so the injectivity radius in K(M_{n}) is bounded above. Letting Λ denote the limit set of (M, ω), we then have

Λ_{n} → Λ

in the Hausdorff topology by Proposition 4.6, but

H. dim(Λ_{n}) → 1 > H. dim(Λ),

so the dimension is discontinuous.

In terms of the Laplacian, the positive ground state ϕ_{n} ∈ L^{2}(M_{n}) of norm 1 becomes more and more concentrated in Y_{n} as n → ∞, and Y_{n} disappears in the limit. In terms of invariant densities, the canonical density μ_{n} for Γ_{n} becomes concentrated at the cusps of Γ, and any limit ν = lim μ_{n} is purely atomic.

III: δ(Γ) is not continuous in the strong topology. Our last example establishes Theorem 1.3 of the introduction, by showing H. dim(Λ) can jump even for a strongly convergent sequence of Kleinian groups.

Let Γ_{t} be the elementary Kleinian group generated by

γ_{t}(z) = e^{2πit}z + 1,

where t = 0 or Im(t) > 0. Then M_{t} = ℍ^{3}/Γ_{t} is homeomorphic to a solid torus S^{1} × D^{2}. For t ∈ ℍ the fundamental group of M_{t} is generated by a geodesic, while M_{0} has a rank-one cusp.

As we saw in the proof of Theorem 5.1, we have:

1. Γ_{t} → Γ_{0} strongly iff t → 0 horocyclically.

2. If t → 0 along a horocycle in ℍ, then a subsequence of Γ_{t} converges geometrically to a rank-two parabolic group Γ′ ⊃ Γ_{0}.

Indeed, the strong convergence occurs exactly when the complex torus X_{t} = Ω(Γ_{t})/Γ_{t} converges geometrically to an infinite cylinder, and this means t → 0 horocyclically. On the other hand, if t → 0 along a horocycle, then [X_{t}] remains in a compact subset of the moduli space ℍ/SL_{2}(ℤ), and a subsequence converges to a torus X_{0} ≅ ℂ/Γ′.

Now the idea is to use the fact that δ(Γ) = r/2 if Γ ≅ ℤ^{r} is an elementary parabolic group of rank r. The discrepancy between the ranks of Γ_{0} and Γ′ will lead to a discontinuity in δ.

We would also like our example groups to be nonelementary. To this end, for R > 4 let

G_{0} = ⟨z ↦ \frac{z}{Rz + 1}, z ↦ z + 1⟩ .

Then G_{0} ≅ ℤ ∗ ℤ is a Fuchsian group; ℍ/G_{0} is a pair of pants with two cusps and one infinite volume end. (When R = 4 the quotient is the triply-punctured sphere.) Since G_{0} is geometrically finite and its limit set is a proper subset of S_{∞}^{1} = ℝ ∪ {∞}, we have

δ(G_{0}) < 1.

In fact, δ(G_{0}) tends to 1/2 continuously as R → ∞, so for any 0 < ϵ < 1/2 we can choose R > 4 such that

δ(G_{0}) = 1/2 + ϵ.

(See the discussion of Hecke groups in [24]).

We can also think of G_{0} in terms of the Klein-Maskit combination theory. A fundamental domain for z ↦ z/(Rz +1) is the region exterior to the two tangent disks

D = {z : |z ± 1/R| < 1/R}.

Since we assume R > 4, D is properly contained in a fundamental domain

F = {z : | Re z| ≤ 1/2}

F − D 0  2/R 1/2 −1/2 −2/R

Figure 4. A fundamental domain for G_{0}.

for the other generator γ_{0}(z) = z+1 of G_{0}. Thus F −D is a fundamental domain for G_{0} (Figure 4). From this picture one sees G_{0} is discrete and free on its generators, and the manifold N_{0} = ℍ^{3}/G_{0} is isomorphic to a connect sum of two copies of M_{0}.

Now let

G_{t} = ⟨z ↦ \frac{z}{Rz + 1}, z ↦ e^{2πit}z + 1⟩ .

Note that G_{t} ⊃ Γ_{t}. For all t inside a small horoball resting on t = 0, we also have D ⊂ F_{t} for a suitable fundamental domain for Γ_{t}. (This is because the torus Ω(Γ_{t})/Γ_{t} approximates the infinite cylinder Ω(Γ_{0})/Γ_{0}.) Thus G_{t} is also free and discrete, and N_{t} = ℍ^{3}/G_{t} is the connect sum of M_{t} and M_{0}. As in the discussion of Γ_{t}, we have G_{t} → G_{0} strongly iff t → 0 horocyclically (by Theorem 5.1).

3. Any sufficiently small horocycle in ℍ resting on t = 0 contains parameters t such that δ(G_{t}) > 1.

Indeed, by Theorem 7.7 any geometric limit G′ of G_{t} satisfies δ(G′) ≤ lim inf δ(G_{t}), while for t moving along a horocycle (2) above says G′ contains a rank two parabolic subgroup Γ′, and thus δ(G′) > 1 (Corollary 2.2).

4. There is a sequence t_{n} ∈ ℍ such that G_{t_{n}} → G_{0} strongly, but

lim δ(G_{t_{n}}) = 1 > δ(G_{0}) = \frac{1}{2} + ϵ.

To construct t_{n}, simply choose horocycles H_{n} converging to 0 and t_{n} ∈ H_{n} such that δ(G_{t_{n}}) > 1; then t_{n} → 0 horocyclically, so G_{t_{n}} → G_{0} strongly, but δ(G_{0}) = 1/2 + ϵ < 1. It follows that δ(G_{t_{n}}) → 1, since otherwise δ would be continuous by Theorem 7.2.

By (4), δ is discontinuous in the strong topology.

Sharpness. This jump in dimension from 1 down to 1/2+ϵ is essentially sharp. Indeed, suppose G_{0} ⊂ Isom(ℍ^{3}) is geometrically finite and G_{n} → G_{0} strongly. If lim δ(G_{n}) > 1, then δ(G_{n}) → δ(G_{0}) by Theorem 7.2; and if δ(G_{0}) ≤ 1/2, then G_{0} is convex cocompact (Corollary 2.2), so we have continuity by Theorem 7.1.

9. Quasifuchsian groups

The Teichmüller space of a surface S leads, via Bers embedding, to many examples of Kleinian groups with interesting algebraic and geometric limits. In this section we study the dimension of the limit set Λ(X, Y) of the quasifuchsian group obtained by gluing together the universals covers of two surfaces X and Y in the Teichmüller space of S.

Dimension as a function on Teichmüller space. We begin by recalling some facts from Teichmüller theory [7], [23, §3]. Let S be a connected compact oriented surface with χ(S) < 0, and let Teich(S) denote the Teichmüller space of Riemann surfaces X marked by S. The Teichmüller metric is defined by

d(X, X′) = \frac{1}{2} inf log K(ϕ),

where ϕ : X → X′ ranges over all quasiconformal mappings compatible with markings, and K(ϕ) ≥ 1 denotes the quasiconformal dilatation of ϕ.

Let AH(S) denote the discrete faithful representations of π_{1}(S) into Isom^{+}(ℍ^{3}), modulo conjugacy, equipped with the topology of algebraic convergence. One can also think of AH(S) as the space of complete hyperbolic 3-manifolds M homotopy equivalent to S.

Let \overline{S} denote S with its orientation reversed. For any pair of Riemann surfaces

(X, Y) ∈ Teich(S) × Teich(\overline{S})

we can construct a quasifuchsian manifold

Q(X, Y) = ℍ^{3}/Γ(X, Y)

marked by S×[0, 1] and hence residing in AH(S). The limit set Λ(X, Y) is a quasicircle, and the domain of discontinuity satisfies

Ω(X, Y)/Γ(X, Y) ≅ X ⊔ Y

with markings respected.

Proposition 9.1. The function H. dim Λ(X, Y) is uniformly Lipschitz on Teich(S) × Teich(\overline{S}).

Proof. Let K = e^{2t} where t = max(d(X_{1}, X_{2}), d(Y_{1}, Y_{2})). Then there is a K-quasiconformal conjugacy between Γ(X_{1}, Y_{1}) and Γ(X_{2}, Y_{2}), and hence a K-quasiconformal map between their limit sets Λ_{1} and Λ_{2}, where K = e^{2t}. Since K-quasiconformal maps are 1/K-Hölder continuous [1, III.C], by general properties of Hausdorff dimension we have

\frac{1}{K} H. dim(Λ_{2}) ≤ H. dim(Λ_{1}) ≤ K H. dim(Λ_{2}),

and therefore

| H. dim(Λ_{1}) − H. dim(Λ_{2})| ≤ 2(K − 1) ≤ 4t + O(t^{2}),

using the fact that both dimensions are ≤ 2. Thus H. dim Λ(X, Y) is Lipschitz with constant 4. □

Sharper estimates can be obtained using [4].

Iteration on a Bers slice. The subspace

B_{Y} = {Q(X, Y) : X ∈ Teich(S)} ⊂ QF (S)

is a Bers slice of quasifuchsian space; it gives a complex-analytic model for Teich(S) as a space of Kleinian groups.

Each element ϕ in the mapping class group Mod(S) determines an automorphism of B_{Y} by sending Q(X, Y) to Q(ϕ(X), Y). It is a fundamental fact that a Bers slice B_{Y} has compact closure in AH(S) [7], so it is interesting to investigate the behavior of Q(ϕ^{n}(X), Y) as n → ∞.

We begin with the case of Dehn twists. A partition P = {C_{1}, …, C_{r}} on S is a collection of isotopy classes of essential disjoint simple closed curves, with no two parallel and none parallel to ∂S. Let M_{P}(X, Y) denote the unique geometrically finite hyperbolic 3-manifold with

M_{P}(X, Y) ≅ int (S × [0, 1] − ⋃_{P} C_{i} × {1/2})

as a topological space, with cusps of rank 1 along ∂S ×[0, 1] and of rank 2 along ⋃ C_{i}×{1/2}, and with conformal boundary X⊔Y corresponding to S × {0, 1}.

By [20] and [11], [12] we have:

Theorem 9.2. Let τ ∈ Mod(S) be a product of Dehn twists

τ = τ_{C_{1}}^{p_{1}} ⋯ τ_{C_{r}}^{p_{r}}

about the curves in a partition P, with each p_{i} ≠ 0. Then as n → ∞,

M_{n} = Q(τ^{n}(X), Y)

converges to algebraic and geometric limits M_{A} and M_{G}, with

M_{G} ≅ M_{P}(X, Y)

and with M_{A} the covering space of M_{G} corresponding to π_{1}(Y).

More precisely, there exists a choice of baseframes ω_{n} ∈ M_{n} such that algebraic and geometric convergence as above is obtained.

We can now apply the results of §7 to arrive at:

Theorem 9.3. For any partition P ≠ ∅ and product of Dehn twists τ as above, the limit sets satisfy

H. dim(Λ_{A}) < H. dim(Λ_{G}) = lim H. dim(Λ(τ^{n}(X), Y)) < 2.

Proof. The algebraic and geometric limits M_{A} and M_{G} both exist, and M_{G} is geometrically finite, so M_{n} → M_{G} strongly by Lemma 4.4. Thus H. dim(Λ_{n}) → H. dim(Λ_{G}) by Theorem 7.2, and H. dim(Λ_{G}) < 2 since the limit set is not the whole sphere.

To show the dimension of Λ_{A} is strictly less than that of Λ_{G}, first note that M_{A} is also geometrically finite. If the dimensions of the limit sets were to agree, then the canonical measures would satisfy μ_{A} = μ_{G} by Theorem 3.1, since there is a unique normalized Γ_{A}-invariant density in the critical dimension δ(Γ_{A}). But then the supports of the canonical measures would coincide, which is impossible since Λ_{A} ≠ Λ_{G}. □

Example. Figure 5 shows the limit sets Λ_{A} and Λ_{G} for τ a single Dehn twist on a surface of genus two. The parameters for this example were computed by Jeff Brock [11], [12].

Figure 5. Algebraic and geometric limits.

Corollary 9.4. For any mapping class ϕ ∈ Mod(S), there is an i > 0 such that

δ(X, Y) = lim_{n→∞} H. dim(Λ(ϕ^{ni}(X), Y))

exists for every (X, Y), and either

1. ϕ^{i} is a product of disjoint Dehn twists, and δ(X, Y) < 2, or

2. ϕ^{i} is pseudo-Anosov on a subsurface, and δ(X, Y) = 2.

Proof. The fact that an iterate ϕ^{i} is either a (possibly trivial) product of Dehn twists, or pseudo-Anosov on a subsurface, follows from Thurston’s classification of surface diffeomorphisms [18], [36]. The first case is handled by the preceding result. For the second (pseudo-Anosov) case, by [11], [12], i can be chosen so the manifolds Q(ϕ^{ni}(X), Y) converge geometrically to a topologically tame, geometrically infinite manifold M_{G}, and hence δ = 2 by Corollary 7.8. □

Pinching a geodesic. Next we investigate approach to infinity by pinching the curves in a partition P.

Theorem 9.5. Suppose X_{n} → ∞ in Teich(S) and for all simple geodesics C we have

ℓ_{C}(X_{n}) → L(C) ∈ [0, ∞].

Let P be the partition consisting of all C with L(C) = 0, and suppose L(C) < ∞ if C is disjoint from ⋃ P. Then there is a geometrically finite manifold M_{A} such that

1. Q(X_{n}, Y) → M_{A} strongly;

2. P is the set of all accidental parabolics on M_{A};

3. All accidental parabolics converge radially; and

4. H. dim Λ(X_{n}, Y) → H. dim Λ_{A}.

Proof. By the conditions on ℓ_{C}(X_{n}), the sequence X_{n} tends to a limit Z in the Deligne-Mumford compactification of the moduli space of S. Here Z is a ‘surface with nodes’ naturally marked by

T = S − ⋃ P;

it has cusps along the components of P, while its geometry outside P is fixed by the limiting lengths of curves C ∉ P. The convergence is compatible with marking outside P. (Compare [19, Appendix A].)

Pass to any subsequence such that the algebraic limit M_{A} ∈ AH(S) exists. We claim M_{A} is the unique geometrically finite manifold with rank-1 cusps along P and with marked conformal boundary

∂M_{A} = Ω(Γ_{A})/Γ_{A} = Z ⊔ Y.

To see this, first note that M_{A} has a cusp at each C ∈ P because

ℓ_{C}(M_{A}) = lim ℓ_{C}(Q(X_{n}, Y)) ≤ 2 lim ℓ_{C}(X_{n}) = 0

(by Bers’ inequality, cf. [7], [22, §6.3]). Next we show ∂M_{A} = Z ⊔ Y. Consider on T any pair of simple curves C and D with positive geometric intersection number. The geodesic representative γ(C) intersects the ruled cylinder D × I between the representatives of D on the two faces of the convex core of Q(X_{n}, Y). Since ℓ_{D}(X_{n}) and ℓ_{D}(Y) are bounded, γ(C) meets an essential loop δ(D) ⊂ D × I of bounded length such that ⟨γ(C), δ(D)⟩ ⊂ π_{1}(Q(X_{n}, Y)) is a nonelementary group. By the Margulis lemma, the length of γ(C) is bounded below, and hence the two faces of the convex hull of Q(X_{n}, Y) are a bounded distance apart (independent of n) along C. It follows that T persists as a subsurface of the conformal boundary of the algebraic limit, and thus Z ⊔ Y ⊂ ∂M_{A}. But the conformal boundary can be no larger than this by area considerations.

We conclude that M_{A} is the geometrically finite manifold described above. Now for each C ∈ P, the quotient torus for π_{1}(C) ⊂ Γ(X_{n}, Y) becomes long and thin as n → ∞, since it contains the π_{1}(C)-covering space of X_{n}, itself an annulus of large modulus. Thus each accidental parabolic converges horocyclically, so Q(X_{n}, Y) → M_{A} strongly by Theorem 5.1 (or by inspection). Since the limit set is connected, we have H. dim(Λ_{A}) ≥ 1; therefore H. dim(Λ(X_{n}, Y) → H. dim(Λ_{A}) by Corollary 7.6. □

Fenchel-Nielsen coordinates. To construct a sequence X_{n} as above, let P be a maximal partition of S. Then one obtains an isomorphism

Teich(S) ≅ ℝ_{+}^{P} × ℝ^{P}

by the map

X ↦ (ℓ_{C}(X), τ_{C}(X))

sending a surface to its Fenchel-Nielsen length and twist parameters (see, e.g. [19]). Consider any sequence of these coordinates (L_{n}(C), T_{n}(C)), C ∈ P, tending to limiting values (L(C), T (C)) where some L(C) = 0. Then it is easy to see the corresponding Riemann surfaces X_{n} satisfy the hypotheses of the Theorem above. The manifolds Q(X_{n}, Y) tend to M_{A} strongly, where ∂M_{A} = Z ⊔ Y and Z is a surface with nodes obtained by gluing together (possibly degenerate) pairs of pants with the limiting length and twist parameters (L(C), T (C)).

In fact one need only require that T_{n}(C) converge for those C ∈ P with L(C) > 0, since twists along the accidental parabolics have no effect on Z. With this proviso, any sequence X_{n} satisfying the Theorem arises via the construction above.

The Fenchel-Nielsen twist. Finally we describe some interesting periodic behavior that occurs for the continuous version of a Dehn twist. Fix a basepoint X_{0} ∈ Teich(S) and a simple closed geodesic C on X_{0} of length L. Define X_{t} by cutting along C, twisting distance tL to the right, and regluing. (In terms of Fenchel-Nielsen coordinates, this means only the twist parameter τ_{C}(X) is varied.) The resulting continuous path in Teichmüller space represents a Fenchel-Nielsen twist deformation of X_{0}.

By construction

X_{t+1} = τ(X_{t}),

where τ ∈ Mod(S) is a right Dehn twist about C. Thus the geometric limit

M_{G}(t) = lim_{n→∞} Q(X_{t+n}, Y) = lim Q(τ^{n}(X_{t}), Y) = M_{P}(X_{t}, Y)

satisfies M_{G}(t + 1) = M_{G}(t); here P = {C}. Let δ(t) denote the Hausdorff dimension of the limit set of M_{G}(t).

Theorem 9.6. The function δ(t) is continuous, δ(t + 1) = δ(t) and

lim_{t→∞} | H. dim(Λ(X_{t}, Y)) − δ(t)| = 0.

Proof. By definition, δ(t + 1) = δ(t). To see δ is continuous, first note that by Theorem 9.3 we have

δ(t) = lim_{n→∞} f(t + n), (9.1)

where f(t) = H. dim Λ(X_{t}, Y). The map t ↦ X_{t} is uniformly continuous in the Teichmüller metric, because X_{t+1} = τ(X_{t}) and τ is an isometry; since H. dim Λ(X, Y) is Lipschitz, f(t) is also uniformly continuous. Thus (9.1) converges uniformly for t ∈ [0, 1]; therefore δ(t) is continuous and the Theorem follows. □

Now recall we have Q(X_{t}, Y) → M_{A} algebraically as t → ∞. If δ(t) is nonconstant (as seems likely), then the Hausdorff dimension of the limit set of Q(X, Y) oscillates like sin(1/x) as Q(X, Y) converges to M_{A} along a Fenchel-Nielsen horocycle.

See [17] for similar results on the dimension of the Julia set of z^{2} + c as c → 1/4.

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Harvard University