j. differential geometry 57 (2001) 173-193

CRITICAL EXPONENT AND DISPLACEMENT OF NEGATIVELY CURVED FREE GROUPS

YONG HOU

Abstract

We study the action of the fundamental group Γ of a negatively curved 3manifold M on the universal cover M̃ of M. In particular we consider the ergodicity properties of the action and the distances by which points of M̃ are displaced by elements of Γ. First we prove a displacement estimate for a general n-dimensional manifold with negatively pinched curvature and free fundamental group. This estimate is given in terms of the critical exponent D of the Poincaré series for Γ. For the case in which n = 3, assuming that Γ is free of rank k ≥ 2, that the limit set of Γ has positive 2-dimensional Hausdorff measure, that D = 2 and that the Poincaré series diverges at the exponent 2, we prove a displacement estimate for Γ which is identical to the one given by the log(2k − 1) theorem [1] for the constant-curvature case.

1. Introduction

In the following M is a complete Riemannian n-manifold with finitely generated fundamental group Γ. We will assume that the sectional curvature K satisfies −b^{2} ≤ K ≤ −a^{2} for some 0 < a ≤ b. A manifold which satisfies curvature bounds of this type will be said to have negatively pinched curvature. The Riemannian universal cover of M is denoted by M̃, and M is identified with M̃/Γ. The following additional notations and terminologies will also be used:

• D is the critical exponent of the Poincaré series

∑_{γ∈Γ} exp(−s dist(x, γx))

of Γ. This means that for every x ∈ M̃, the series diverges when s < D and converges when s > D.

Received February 22, 2001.

• Γ is said to be divergent if the Poincaré series diverges at s = D.

• S denotes a arbitrary, fixed free generating set of Γ, with convention that the inverses are not included.

• Λ(Γ) is the limit set of Γ, which is the unique minimal closed Γ-invariant subset of S_{∞}.

• 𝔇 denotes the Hausdorff dimension of Λ(Γ) (with respect to the Busemann metric, the Gromov metric or the shadow metric: See §2).

• 𝔐^{d} denotes the d-dimensional Hausdorff measure on S_{∞} with respect to the Busemann metric.

• For each x ∈ M̃ and γ ∈ Γ, we refer to dist(x, γx) as the displacement of x under γ.

Theorem 1.1. Suppose that −1 ≤ K ≤ −a^{2} and that Γ is free. Then

∑_{γ∈S} \frac{1}{1 + exp(D dist(x, γx))} ≤ \frac{1}{2}.

Theorem 1.2. Let M = M̃/Γ be a 3-manifold with −1 ≤ K ≤ −a^{2}. Suppose that Γ is free, that 𝔐^{2}(Λ(Γ)) > 0, that D = 2 and that Γ is divergent. Then the displacement satisfies

∑_{γ∈S} \frac{1}{1 + exp(dist(x, γx))} ≤ \frac{1}{2}.

As corollary to the above Theorem 1.1, we have the following

Corollary 1.3. Suppose that M is a rank-1 locally symmetric space normalized so that −1 ≤ K ≤ −a^{2}, and that Γ is free. Then,

∑_{γ∈S} \frac{1}{1 + exp(𝔇 dist(x, γx))} ≤ \frac{1}{2}.

The study of the displacement function of fundamental group Γ is crucial in understanding the geometry of the manifold M. In constant curvature spaces, the current best estimate is due to the work of Culler and Shalen [4] where they have obtained an log 3 lower bound for the displacement of rank 2 free group. Later the same estimate has been generalized to rank k in [1] where the log(2k − 1) theorem is obtained.

In this paper we prove the estimate given by Theorem 1.1. The crucial difference from the constant curvature case is the involvement of the critical exponent in the estimate. Hence it provides an relationship between displacement and Hausdorff dimension 𝔇. When the critical exponent D is < 1 we have a estimate which is stronger than the log(2k − 1) estimate even for constant curvature spaces (see Corollary 3.7). Theorem 1.2 is closely related to the question of rigidity for Γ with D = 2 (see Theorem 3.5). In [8] a topological condition on M has been established for which Γ is divergent at D.

In Section 2 will study invariant densities on S_{∞} and in particular a decomposition theorem of Γ with respect to the conformal invariant density is proved. Section 3 is devoted to proving the displacement function estimates and additional corollaries.

Acknowledgements

I would like to express my gratitude to Peter Shalen for his constant encouragment and support. I also think Marc Culler for his interest. I am also grateful to Sun Su Lan and Shu Ying Hou for their continuous support and guidance. I am deeply indebted to my brothers Qun Hou and Wei Hou for their inspiration and dedication.

2. The boundary S_{∞}

In some situations we will take the dimension of M to be 3, otherwise we will assume M is n-dimensional in general.

Let y ∈ M̃ and ζ ∈ S_{∞} be given. The following notations will be assumed throughout.

• S_{y} denotes the unit sphere in the tangent space at y.

• Φ_{y} denotes the natural homeomorphism between S_{∞} and S_{y}.

• c_{y}^{ζ}(t) denotes the geodesic ray connecting y and ζ.

2.1 Metrics on S_{∞}

Fix a point x ∈ M̃. Let δ > 0 be a positive real number. Let ξ, ζ in S_{∞} be given.

In [7], Gromov defined a metric on S_{∞} as follows. Let y, z ∈ M̃ be given. Let us consider arbitrary continuous curve c(t) in M̃ with initial point and end point denoted by c(t_{0}) = y and c(t_{1}) = z respectively. Define a nonnegative real-valued function G_{x} on M̃ × M̃ by

G_{x}(y, z) := inf_{all c} (∫_{[t_{0},t_{1}]} exp(−δ dist(x, c(t))) dt) .

In particular, Gromov showed that there exists δ(b) > 0 depending only on the lower pinching constant b such that for any δ with 0 < δ ≤ δ(b), the function G_{x} extends continuously to S_{∞}×S_{∞} and defines a distance. Every element of Γ extends to S_{∞} as a Lipschitz map with respect to G_{x}.

There are also many other equivalent metrics on S_{∞}. In particular, Kaimanovich [9] has studied the following metrics.

K_{x} metric : Let B_{ζ} denote the Busemann function based at x_{0}, i.e.,

B_{ζ} := lim_{τ→∞} dist(x, c_{x_{0}}^{ζ}(τ)) − τ.

Changing the base point x_{0} will change B_{ζ} only by adding a constant. Hence the function defined by B_{ζ}(x, y) := B_{ζ}(x)−B_{ζ}(y), for x, y ∈ M̃, is independent of the base point. The function B_{ζ}(x, y) is called the Busemann cocycle. Define a real-valued function β_{x} : S_{∞} × S_{∞} ⟶ ℝ by β_{x}(ξ, ζ) := B_{ξ}(x, y) + B_{ζ}(x, y) where y is a point on the geodesic connecting ξ and ζ. It is clear from the definition of the Busemann function that β_{x}(ξ, ζ) is defined independently of the choice of y. Geometrically, β_{x}(ξ, ζ) is the length of the segment on the geodesic connecting ξ and ζ cut out by the horospheres centered at ξ and ζ passing through the point x. The K_{x} metric is then defined by

K_{x}(ξ, ζ) := exp (−\frac{1}{2}δβ_{x}(ξ, ζ)) .

L_{x} metric: Let α_{x}(ξ, ζ) denote the distance between x and the geodesic connecting ξ and ζ. The function L_{x} : S_{∞} × S_{∞} ⟶ ℝ is then defined by

L_{x}(ξ, ζ) := exp(−δα_{x}(ξ, ζ)).

d_{x} metric: Define a function l_{x} : S_{∞} × S_{∞} ⟶ ℝ by

l_{x}(ξ, ζ) := sup {τ | dist (c_{ξ}^{x}(τ), c_{ζ}^{x}(τ)) = 1} .

Geometrically, a neighborhood about ξ in S_{∞} with respect to the topology induced by l_{x} is the shadow cast by the intersection of 1-ball about c_{ξ}^{x}(τ) and τ-sphere about x. The d_{x} metric is then defined by

d_{x}(ξ, ζ) := exp(−δl_{x}(ξ, ζ)).

Proposition 2.1 ([9]). There exists a positive number δ(a, b) > 0 depending only on the pinching constants a and b such that for every 0 < δ ≤ δ(a, b) the metrics K_{x}, L_{x}, d_{x} are equivalent to Gromov’s metric.

From now on we will fix a δ > 0 having the property stated in Proposition 2.1.

For completeness we include the following comparison theorem by Toponogov [10]. Let M̃_{a} and M̃_{b} denote the simply connected complete constant curvature manifolds with K = −a and K = −b respectively. Denote a geodesic triangle with θ = ∠A and r = dist(A, B), t = dist(A, C), s = dist(B, C) by △ABC.

Theorem 2.2 (Toponogov’s Comparison). Keeping the above notation, let △ABC be a geodesic triangle in M̃. Then we have

(i)  cos θ sinh ar sinh at ≥ cosh ar cosh at − cosh as,
(ii)  cos θ sinh br sinh bt ≤ cosh br cosh bt − cosh bs.

2.2 Γ-Invariant Density on S_{∞}

First, let us recall a simple uniqueness result.

We will say that two Borel measures on S_{∞} are in the same Γ-class if the Radon-Nikodym derivative of γ^{∗}ν_{1} with respect to ν_{1} is equal to the Radon-Nikodym derivative of γ^{∗}ν_{2} with respect to ν_{2}.

Proposition 2.3. Let Γ be nonelementary and discrete. Suppose that Γ acts ergodically on S_{∞} with respect to a measure ν defined on S_{∞}. Then every measure of S_{∞} in the same measure class as ν is a constant multiple of ν.

Proof. Denote the measure class of ν by [ν]. Let μ ∈ [ν] then σ := \frac{1}{2}(μ+ν) is also in [ν]. Since both ν and μ are absolutely continuous with respect to σ, their Radon-Nikodym derivatives with respect to σ defines a ν-measurable Γ-invariant function on S_{∞}. Then, the result follows from ergodicity of Γ, which implies these functions are equal to a non-zero constant almost ν-everywhere. q.e.d.

A subset A of M̃ is called uniformly discrete if there exists a positive number ϵ such that for any two distinct point x, y ∈ A we have dist(x, y) > ϵ.

Definition 2.4. Let V ⊂ M̃ be a uniformly discrete subset. Let h_{s}(x, y) : M̃ × M̃ ⟶ ℝ^{+} be a family of positive continuous functions indexed by s ∈ ℝ such that K_{s}(x, y, ζ) := lim_{z→ζ} \frac{h_{s}(x,z)}{h_{s}(y,z)} exists and

defines a family of continuous functions on M̃× M̃× S_{∞}. Suppose that there exists a number α for which the series Z_{x,s,V} := ∑_{v∈V} h_{s}(x, v) is convergent when s > α and divergent when s ≤ α for any x ∈ M̃. Then we call Z_{s,V} an α-series of V. If we also have h_{s}(γx, γy) = h_{s}(x, y) for every isometry γ of M̃, then Z_{s,V} will be called an invariant α- series of V.

For a given α-series Z_{s,V} of V we will define a Borel measure ν_{x,Z,V,s} on V by ∑_{v∈V} h_{s}(x, v)δ_{v}.

Proposition 2.5. Let x be a point of M̃ and W ⊂ M̃ infinite uniformly discrete subset. Let Z_{s,W} be an α-series of W. Then for every V ⊂ W there exists a sequence (s_{i}) of real numbers larger than α such that:

(a) lim_{i→∞} s_{i} = α.

(b) The sequence (Z_{x,s_{i},W}^{−1} ν_{x,Z,V,s_{i}})_{i} is a weakly convergent sequence of Borel measures of M̃ ∪ S_{∞} with mass at most 1, and the limit Borel measure denoted by ν_{x,Z,V} is supported on S_{∞}.

(c) ν_{x,Z,W} is a probability measure.

(d) [ν_{y,Z,V}]_{y∈M̃} is a density of S_{∞} with Radon-Nikodym derivative at ζ ∈ S_{∞} given by K_{α}(x, y, ζ).

Proof. Take a sequence (s′_{i}) which converges to α from above. For all i, the Borel measure Z_{x,s′_{i},W}^{−1} ν_{x,Z,V,s′_{i}} has mass at most 1 . Hence there exists a subsequence (s′_{i_{j}}) of (s′_{i}) such that Z_{x,s′_{i_{j}} ,W}^{−1} ν_{x,Z,V,s′_{i_{j}}} converges

weakly to a Borel measure ν_{x,Z,V} with mass at most 1. If V = W then the measures in the above sequence are probability measures, hence ν_{x,Z,W} is a probability measure. If B ⊂ M̃ is any compact subset then B ∩ V is finite. Since Z_{x,s′_{i_{j}} ,W} ⟶ ∞, we must have ν_{x,Z,V} (B ∩ V) = 0.

Hence, ν_{x,Z,V} has support contained in S_{∞}. Finally, for y ∈ M̃ and ζ ∈ S_{∞} we have

lim_{z→ζ} \frac{h_{s′_{i_{j}}} (x, z)}{h_{s′_{i_{j}}} (y, z)} = K_{s′_{i_{j}}} (x, y, ζ),

so the Radon-Nikodym derivative of ν_{x,Z,V} is given by the limit of K_{s′_{i_{j}}} (x, y, ζ) as s′_{i_{j}} ⟶ α, which is K_{α}(x, y, ζ). q.e.d.

Proposition 2.6. Let W be an infinite uniformly discrete subset of M̃. Let V be a collection of subsets of W with W ∈ V. Suppose we have an invariant α-series Z_{s,W} of W. Then there exists a family (ν_{V})_{V ∈V} of Borel measures for M̃ ∪ S_{∞}, indexed by the collection V and satisfying the following conditions:

(1) ν_{W} (S_{∞}) = 1.

(2) For any finite collection (V_{j})_{1≤j≤n} of disjoint sets in V with V := ∐_{j} V_{j} ∈ V, we have ν_{V} = ∑_{j} ν_{V_{j}}.

(3) For any V ∈ V and γ isometry of M̃ with γV ∈ V we have γ^{∗}ν_{γV} = ν_{V}.

(4) The support of ν_{V} is contained in V ∩ S_{∞}.

Proof. From Proposition 2.5 (c), we have (1). For (2), let χ_{V_{j}} be the characteristic function of V_{j}. Then ν_{V_{j}} = ν_{V} χ_{V_{j}}, which gives ∑_{j} ν_{V_{j}} = ∑_{j} ν_{V} χ_{V_{j}} = ν_{V}. Let γ be a isometry of M̃. Since, Z_{s,W} is an invariant α-series, we have γ^{∗}ν_{γx,Z,γV,s} = ν_{x,Z,V,s}; therefore by Proposition 2.5 we have γ^{∗}ν_{γx,γV} = ν_{x,V}. This gives us (3). Finally, (4) follows easily from Proposition 2.5 (b). q.e.d.

Next we will prove a result which is a generalization of the Culler-Shalen paradoxical decomposition theorem to this abstract setting.

Theorem 2.7. Let Γ be a finitely generated, free discrete group of isometries of M̃ with generating set Ω. Let x ∈ M̃ be given. Suppose we have an invariant α-series Z_{s,W} with W = Γx. Set Ψ := Ω ∐ Ω^{−1}. Then there exist a Γ-invariant density [μ_{y}]_{y∈M̃} on S_{∞}, and a family [ν_{ψ}]_{ψ∈Ψ} of Borel measures on S_{∞} with:

(1) μ_{x}(S_{∞}) = 1.

(2) μ_{x} = ∑_{ψ∈Ψ} ν_{ψ}.

(3) ∫_{S_{∞}} K(x, ψ^{−1}x, ξ)dν_{ψ^{−1}}(ξ) = 1 − ∫_{S_{∞}} dν_{ψ}.

Proof. Let us write every element γ ∈ Γ as a reduced word ψ_{1} ⋯ ψ_{n} with {ψ_{j}} ⊂ Ψ. Then we have the decomposition of Γ as Γ = {1} ∐ ∐_{ψ∈Ψ} I_{ψ}, where I_{ψ} is the set of nontrivial elements in Γ with inital letter ψ. By the fact that Γ act freely on M̃ we have W = Γx = {x} ∐ ∐_{ψ∈Ψ} V_{ψ} where V_{ψ} = {γx : γ ∈ I_{ψ}}. Let V denote the collection consisting of all sets of the form ∐_{ψ∈Ψ′} V_{ψ} or {x} ∐ ∐_{ψ∈Ψ′} V_{ψ} for Ψ′ ⊂ Ψ. Applying Proposition 2.6 with W and V so defined, we get a family of Borel measures (μ_{y,V_{ψ}})_{y∈M̃} for each ψ ∈ Ψ. By Proposition 2.6 (1), μ_{x,W} is a probability measure on S_{∞}, which gives (1). Define ν_{ψ} := μ_{x,V_{ψ}} for each ψ ∈ Ψ. By the above decomposition of W, we have μ_{x,W} = μ_{x,x} +∑_{ψ∈Ψ} ν_{ψ}. But μ_{x,x} = 0, which gives (2). Since W = V_{ψ} ∐ ψV_{ψ^{−1}}, we have ψV_{ψ^{−1}} ∈ V. Then by (3) of Proposition 2.6, we get

μ_{ψ^{−1}x,V_{ψ^{−1}}} = ψ^{∗}μ_{x,W −V_{ψ}} = ψ^{∗}(μ_{x} − ν_{ψ}).

By Proposition 2.5 (d), we have

dμ_{ψ^{−1}x,V_{ψ^{−1}}} = K(ψ^{−1}x, x, ξ)dμ_{x,V_{ψ^{−1}}} .

From this, we get

∫_{S_{∞}} K(ψ^{−1}x, x, ξ)dν_{ψ^{−1}} = ∫_{S_{∞}} d(ψ^{∗}(μ_{x,W} − ν_{ψ})) = 1 − ∫_{S_{∞}} dν_{ψ}.

The last equality gives us (3), which concludes the proof. q.e.d.

Next we will construct two α-series of W = Γx. The first construction is based on the work of Ancona, uses the λ-Green’s function, which gives rise to the Poisson kernel density (λ-harmonic measure) on S_{∞}. The second construction, utilizes the Poincaré series, gives rise to the Patterson-Sullivan measure [15], which is a D-conformal density.

Harmonic density. Let λ_{1} and λ̃_{1} denote the infinum of the spectrum of Δ on M = M̃/Γ, and of Δ̃ on M̃, respectively. Recall that for a noncompact open manifold, the infinum of the spectrum is characterized as

λ_{1} := inf_{f∈C_{o}^{∞},f≠0} (\frac{∫ |∇f|^{2}}{∫ f^{2}} )

where C_{o}^{∞} is the space of smooth functions on M with compact support. Note that we always have λ_{1} ≤ λ̃_{1}.

The λ_{1}-harmonic functions have been studied by Ancona in [2].

Proposition 2.8 (Ancona). For each s < λ_{1}, the elliptic operator Δ̃ + sI has a Green function G_{s}(x, y), and there exists a function f : ℝ^{+} ⟶ ℝ^{+} such that ∑_{γ∈Γ} Ĝ_{s}(x, γy) converges for s < λ_{1} and diverges for s ≥ λ_{1}, where Ĝ_{s}(x, γy) := exp(f(dist(y, γy)))G_{s}(x, γy). Furthermore, 𝔓_{s}(x, y, ζ) := lim_{z→ζ} \frac{G_{s}(x,z)}{G_{s}(y,z)} defines the Poisson kernel of

Δ̃ + sI at ζ ∈ S_{∞}.

Theorem 2.9. Let x be any point of M̃. With notation as in Proposition 2.8, the series ∑_{v∈V} Ĝ_{s}(x, v) is a λ_{1}-series for W = Γx. Furthermore, there is a family of Borel measures [ω_{y}^{1}]_{y∈M̃} on S_{∞} such that:

(i) For all x, y ∈ M̃, Radon-Nikodym derivative dω_{y}^{1}/dω_{x}^{1} at any point ζ ∈ S_{∞} is equal to 𝔓_{λ_{1}}(x, y, ζ).

(ii) ω_{x}^{1} is of mass 1.

Proof. The first assertion follows from Proposition 2.5 with α = λ_{1}. The second assertion then follows from Proposition 2.8. q.e.d.

λ-dimensional Hausdorff measure. It is not straightforward to define an “area” measure on S_{∞} in the variable curvature case. However, there are useful ways of doing this, which involve making appropriate choices of a metric on S_{∞} and considering the corresponding Hausdorff measure.

For reasons that will become clear shortly, we will be working with the K_{x} metric on S_{∞}. Let C be a subset of S_{∞}. The λ-dimensional Hausdorff measure 𝔐_{x}^{λ}(C) of C on the metric space(S_{∞}, K_{x}) is defined as lim_{r→0} 𝔐_{r}^{λ}(C) where

𝔐_{r}^{λ} := inf{∑_{j} r_{j}^{λ/δ} | C ⊂ ∪_{j}B(ξ_{j}, r_{j}); ξ ∈ C, r_{j} ≤ r}.

Here B(ξ_{j}, r_{j}) is an r_{j}-ball about ξ_{j} with respect to the metric K_{x}. Observe that for any x ∈ M̃ and any γ ∈ Γ, we have γ^{∗}𝔐_{x}^{λ} = 𝔐_{γ^{∗}x}^{λ}; this follows from the straightforward identity B_{γζ}(γx, γy) = B_{ζ}(x, y).

(Note that in the case of ℍ^{3}, if we take a round metric about a point z_{0} in ℍ^{3} and denote its extension to ∂ℍ^{3} by ρ_{0}, then 𝔐_{z_{0}}^{2} is the area metric corresponding to the spherical metric A_{z_{0}}.)

A family of finite Borel measures [ν_{y}]_{y∈M̃} will be called a λ-confor-

mal density under the action of Γ if for every x ∈ M̃ and every γ ∈ Γ we have γ^{∗}ν_{y} = ν_{γ^{∗}y}, and the Radon-Nikodym derivative \frac{dν_{y}}{dγ^{∗}ν_{y}} (ζ) at any point ζ ∈ S_{∞} is equal to exp(−λB_{ζ}(γ^{−1}y, y)). (This is to be interpreted as being vacuously true if, for example, the measures in the family are all identically zero.)

First we recall a fundamental fact about conformal density, which was originally proved by Sullivan in the hyperbolic case and generalized to the pinched negatively curved spaces by Yue [16]. It relates the divergence of Γ at the critical exponent D with ergodicity of the Dconformal density under the action of Γ.

Proposition 2.10 (Sullivan). Let Γ be a nonelementary, discrete, torsion-free and divergent at D. Suppose [ν] is a D-conformal density under the action of Γ, then Γ is ergodic with respect to [ν].

Proposition 2.11. Let Γ be a discrete group of isometries of M̃. Suppose 𝔐_{x}^{λ} is a finite measure. Then 𝔐_{x}^{λ} is a λ-conformal density under the action of Γ.

Proof. Let γ ∈ Γ and ξ, ζ ∈ S_{∞} be given. By definition we have β_{x}(γξ, γζ) = B_{γξ}(x, γy)+B_{γζ}(x, γy) where γy is a point on the geodesic connecting γξ and γζ. Since

B_{γξ}(x, γy) = B_{γξ}(x) − B_{γξ}(γy)
= B_{ξ}(γ^{−1}x) − B_{ξ}(y)

we have

β_{x}(γξ, γζ) = [B_{ξ}(γ^{−1}x) − B_{ξ}(y) + B_{ξ}(x) − B_{ξ}(x)]
+[B_{ζ}(γ^{−1}x) − B_{ζ}(y) + B_{ζ}(x) − B_{ζ}(x)]
= B_{ξ}(γ^{−1}x, x) + B_{ζ}(γ^{−1}x, x) + β_{x}(ξ, ζ).

Hence, as ξ ⟶ ζ we have B_{ξ}(γ^{−1}x, x) + B_{ζ}(γ^{−1}x, x) ⟶ 2B_{ζ}(γ^{−1}x, x).

Therefore

lim_{ξ→ζ} \frac{K_{x}(γξ, γζ)}{K_{x}(ξ, ζ)} = exp(−δB_{ζ}(γ^{−1}x, x)).

Finally, using the fact that γ^{∗}𝔐_{x}^{λ} = 𝔐_{γ^{∗}x}^{λ} and the above transformation property of K_{x} we have the desired result :

\frac{d𝔐_{x}^{λ}}{dγ^{∗}𝔐_{x}^{λ}} (ζ) = exp(−λB_{ζ}(γ^{−1}x, x)).

q.e.d.

Proposition 2.12. Suppose dim M̃ = 3. Let A_{x} be the normalized area measure on S_{x}. Then, for any Borel subset C ⊂ S_{x} we have

Φ_{x}^{∗}𝔐_{x}^{2b}(C) ≤ A_{x}(C).

Hence, if 𝔐_{x}^{2b}(Λ(Γ)) > 0 then by normalization, 𝔐_{x}^{2b} defines a probability measure on Λ(Γ) which is bounded by A_{x}.

Proof. Let ξ and ζ in S_{∞}. Then we have

B_{x}(ξ, ζ) = lim_{τ→∞}(2τ − dist(y, c_{x}^{ξ}(τ)) − dist(y, c_{x}^{ζ}(τ)))

where y is any arbitrary point on the geodesic connecting ξ and ζ. By letting y ⟶ ξ we get β_{x}(ξ, ζ) = lim_{τ→∞}(2τ − dist(c_{x}^{ξ}(τ), c_{x}^{ζ}(τ)).

Let B_{K}(ζ, r) denote the ball of radius r about ζ with respect to the K_{x}-metric. Let ξ ∈ B_{K}(ζ, r) be an element with exp(−bβ_{x}(ξ, ζ)) = r^{2b/δ} and minimal ∠xξζ. By the above formula for β_{x}(ξ, ζ) we then have K_{x}(ξ, ζ) = lim_{τ→∞} exp(\frac{δ}{2}(dist(c_{x}^{ξ}(τ), c_{x}^{ζ}(τ)) − τ)) . Let s(τ) denote dist(c_{x}^{ξ}(τ), c_{x}^{ζ}(τ)) and θ := ∠xξζ. Then using inequality (i) of Proposition 2.2 we have:

cosh^{2} bτ − cosh bs(τ) ≥ cos θ sinh^{2} bτ.

Substituting 1 + sinh^{2} bτ for cosh^{2} bτ we get

sinh^{2} bτ(1 − cos θ) + 1 ≥ cosh bs(τ).

By using the equality sinh^{2} \frac{bs(τ)}{2} = \frac{1}{2}(cosh bs(τ) − 1) we then obtain

sinh^{2} bτ(1 − cos θ) ≥ 2 sinh^{2} \frac{bs(τ)}{2},

or

\{\frac{exp(bs(τ)/2) − exp(−bs(τ)/2)}{exp(bτ) − exp^{(} −bτ)}\}^{2} ≤ \frac{1}{2}(1 − cos θ).

Hence for large τ we have

lim_{τ→∞}\{exp(bs(τ)/2 − bτ)\}^{2} ≤ \frac{1}{2}(1 − cos θ).

By using the last equation we get r^{2b/δ} ≤ A_{x}(Φ_{x}^{−1}(B_{K}(ζ, r))).

Denote by C′ := Φ_{x}(C). Let B := ∪_{j}B_{K}(ζ_{j}, r_{j}) be a cover of C′, with ζ_{j} ∈ C′ and r_{j} ≤ r. Then

𝔐_{r}^{2b}(C′) = inf_{B} ∑_{j} r_{j}^{2b/δ} ≤ inf_{B} ∑_{j} A_{x}(Φ_{x}^{−1}(B_{K}(ζ_{j}, r_{j})).

Let V_{j} := Φ_{x}^{−1}B_{K}(ζ_{j}, r_{j}). Then C ⊂ ∪_{j}V_{j} is a cover of C. Hence we have inf_{∪V_{j}} ∑ A_{x}(V_{j}) = A_{x}(C) by regularity of A_{x}. Therefore 𝔐_{r}^{2b}(C′) ≤ A_{x}(C). Letting r → 0 we get the desired result. q.e.d.

Corollary 2.13. Suppose 𝔐_{x}^{2b}(Λ(Γ)) > 0. Then [𝔐_{y}^{2b}]_{y∈M̃} can be normalized so as to define a 2b-conformal density under the action of Γ whose total mass is 1.

Proof. The corollary follows from Proposition 2.11 and Proposition 2.12. q.e.d.

The Patterson-Sullivan construction. Let x ∈ M̃ and s > 0 be given. Denote the Poincaré series for a infinite uniformly discrete subset W ⊂ M̃ by Z_{W} (x, s), i.e., set

Z_{W} (x, s) := ∑_{v∈W} exp(−s dist(x, v)).

In particular, W can be the orbit Γx of x. We will use the notation Z_{Γ}(x) to denote the Poincare series for W = Γx in this case.

Proposition 2.14. There is a unique number 0 ≤ D ≤ (n − 1)b, such that for s > D, Z_{W} converges and Z_{W} diverges for s < D.

If Z_{W} (x, D) is divergent, then we say that Γ is divergent.

Let V ⊂ W. First consider the case that Z_{V} (x, D) is divergent. Following Patterson-Sullivan, we will construct a family of Borel measures. Define:

μ_{V,x,s} = \frac{1}{Z_{V} (x, s)} ∑_{v∈V} exp(−s dist(x, v))δ_{v} ; s > D.

Since Z_{V} (x, D) = ∞, we have as s ⟶ D through a suitable sequence, μ_{V,x,s} converges weakly to a limit probability measure, which we denote by μ_{V,x}. This limit measure has its support contained in S_{∞}.

From the definition of μ_{V,x,s} we have

\frac{dμ_{V,x,s}}{dμ_{V,y,s}} (v) = exp(s(dist(y, v) − dist(x, v))).

As v ⟶ ζ, we have exp(s(dist(y, v) − dist(x, v))) ⟶ exp(−sB_{ζ}(x, y)). Hence

\frac{dμ_{V,x,s}}{dμ_{V,y,s}} (v) ⟶ \frac{dμ_{V,x}}{dμ_{V,y}} (ζ) = exp(−DB_{ζ}(x, y)).

Therefore [μ_{V,x}] is a D-conformal density.

The above construction was based on the assumption that Z_{V} (x, D) = ∞. If Z_{V} (x, D) < ∞ we can use the following lemma proved by Patterson [12] and presented in [4].

Lemma 2.15. There exits a real-valued function α(t), such that the perturbed Poincaré series Z̃_{V} (x, s) := ∑_{v∈V} exp(−α(dist(x, v))) is finite for s > D and infinite when 0 ≤ s ≤ D.

Proof. Let s_{k} := θ_{k}D for some increasing sequence of positive numbers θ_{k} ⟶ 1. Let {R_{k}}_{k≥1} be a monotone increasing sequence of positive numbers with R_{k} ⟶ ∞ and dist(x, v) ≤ R_{k} for all v ∈ V_{k}. We also choose V_{k} ⊂ V so that

∑_{v∈V_{k}} exp(−s_{k} dist(x, v)) ≥ k.

Note that 0 < s_{k} < D, so Z_{V} (x, s_{k}) < ∞. Let β(t) denote a continuous increasing function with β(R_{k}) = θ_{k} and β(t) < 1 for t ≥ 0. Then the desired adjustment function α is defined by α(t) := ∫_{0}^{t} β(τ)dτ ≤ θ_{k}t for 0 ≤ t ≤ R_{k}. To see this, note that

Z̃(x, D) ≥ ∑_{v∈V_{k}} exp(−Dα(dist(x, v))) ≥ ∑_{v∈V_{k}} exp(Ds_{k} dist(x, v)) ≥ k

for every k ≥ 1. So we have Z̃(x, D) = ∞. q.e.d.

Note that, Z̃(x, D) = ∞ for one x implies Z̃(x, D) is divergent for all x, by the fact that Z̃(x′, s) ≤ exp(s dist(x, x′))Z̃(x, s), for all x′, x ∈ M̃.

Proposition 2.16. For a given adjustment function α, we have:

exp(α(dist(x′, v)) − α(dist(x, v)))⟶ exp(B_{ξ}(x′, x))

as v → ξ. Hence the above construction with Z̃_{V} (x, s) in place of Z̃_{V} (x, s) still defines a conformal density.

Proof. Let {v_{k}} be a sequence in V converges to ξ ∈ S_{∞}. Then dist(x, v_{k}) ⟶ ∞ and

lim_{k→∞} exp(dist(y, v_{k}) − dist(x, v_{k})) = exp(B_{ξ}(y, x)).

Since α′(t) = β(t) approaches to 1 as t ⟶ ∞, we have

lim_{k→∞} \frac{α(dist(y, v_{k})) − α(dist(x, v_{k}))}{dist(y, v_{k}) − dist(x, v_{k})} = 1,

which gives the desired result:

lim_{k→∞} exp(α(dist(y, v_{k})) − α(dist(x, v_{k}))) = exp(B_{ξ}(y, x)).

q.e.d.

Proposition 2.17. The series Z̃_{V} (x, s) defined by Lemma 2.15 is an invariant D-series for V with D ∈ [0, (n − 1)b], and its Radon-Nikodym derivative at ζ ∈ S_{∞} is given by exp(−DB_{ζ}(x′, x)).

Proof. The proposition follows from Proposition 2.14, Lemma 2.15 and Proposition 2.16. q.e.d.

Let us define a function Θ : M̃ × M̃ × S_{∞} ⟶ ℝ^{+} by Θ(x, y, ξ) := exp(−B_{ξ}(x, y)).

Theorem 2.18. Let Γ be a finitely generated, free discrete group of isometries of M̃ with free generating set Ω. Let x ∈ M̃. Set Ψ := Ω ∐ Ω^{−1}. Then there exist a D-conformal density [μ_{y}]_{y∈M̃}^{D} under the action of Γ, and a family [ν_{ψ}]_{ψ∈Ψ} of Borel measures on S_{∞} with:

(1) μ_{x}(S_{∞}) = 1.

(2) μ_{x} = ∑_{ψ∈Ψ} ν_{ψ}.

(3) ∫_{S_{∞}} Θ^{D}(x, ψ^{−1}x, ξ)dν_{ψ^{−1}}(ξ) = 1 − ∫_{S_{∞}} dν_{ψ}.

Proof. This follows from Proposition 2.17 and Theorem 2.7. q.e.d.

In the case where the curvature is a constant k, the density of Patterson-Sullivan conformal measure coincides with the Poisson kernel. Let P_{k} denote the corresponding Poisson kernel on M̃_{k}. Then, P_{k} : M̃ × M̃ × S_{∞} ⟶ ℝ is given by

P_{k}(x, y, ξ) = (cosh k dist(x, y) − sinh k dist(x, y) cos ∠yxξ)^{−1}.

Proposition 2.19. Let (x, y, ξ) ∈ M̃×M̃×S_{∞} and λ ≥ 0 be given. Then we have

Θ^{aλ}(x, y, ξ) ≥ P_{a}^{λ}(x, y, ξ) (I)

Θ^{bλ}(x, y, ξ) ≤ P_{b}^{λ}(x, y, ξ) (II)

Proof. Since

exp(−aλB_{ξ}(x, y)) = lim_{v→ξ} \frac{cosh aλ dist(y, v)}{cosh aλ dist(x, v)}

and by Proposition 2.2 (i) we have

\frac{cosh aλ dist(y, v)}{cosh aλ dist(x, v)}

≥ (cosh aλ dist(x, y) − tanh aλ dist(y, v) sinh aλ dist(x, y) cos ∠xyv)^{−1}.

Hence, by letting v ⟶ ξ we have exp(−aλB_{ξ}(x, y)) ≤ P_{a}^{λ}(x, y, ξ), which proves (I). The proof for (II) is similar, but we use Proposition 2.2 (ii) in place of 2.2 (i). q.e.d.

3. Displacement

In this section, we will use the previous results to study the displacement function of Γ. We set b = 1 ( i.e −1 ≤ K ≤ −a^{2}) throughout this section.

3.1 Useful lemmas

Lemma 3.1. Let a point x ∈ M̃ and an isometry γ of M̃ be given. Let α and β be nonnegative numbers with α ≤ 1/2 and β ≤ 1. Let λ ≠ 0 be a positive number. Suppose that there exists a Borel measure ν on S_{∞} such that:

188 yong hou

(i) ν(S_{∞}) ≤ α. (ii) ∫_{S_{∞}} Θ(x, γ^{−1}x, ξ)^{λ}dν ≥ β. Then

dist(x, γx) ≥ \frac{1}{λ} log \frac{β}{α}.

Proof. By Proposition 2.19 we have

β ≤ ∫_{S_{∞}} Θ(x, γ^{−1}x, ξ)^{λ}dν ≤ ∫_{S_{∞}} P_{1}^{λ}(x, γ^{−1}x, ξ)dν
≤ sup_{S_{∞}} P_{1}^{λ}ν(S_{∞})
≤ (\frac{1}{cosh(dist(γx, x)) − sinh(dist(γx, x))})^{λ} α.

Using the last inequality and the definitions of cosh z and sinh z, we have

\frac{β}{α} ≤ exp(λ dist(γx, x)),

which gives

dist(x, γx) ≥ \frac{1}{λ} log \frac{β}{α}.

q.e.d.

Lemma 3.2. Let dim M = 3. Fix a point x ∈ M̃. Let us use the notation of Lemma 3.1. Suppose that ν satisfies:

(i) ν ≤ 𝔐_{x}^{2}. (ii) ν(S_{∞}) ≤ α. (iii) ∫_{S_{∞}} Θ(x, γ^{−1}x, ξ)^{2}dν ≥ β. Then we have

dist(x, γx) ≥ \frac{1}{2} log \frac{β(1 − α)}{α(1 − β)}.

Proof. We define a function f : ℝ × [0, π] ⟶ ℝ by

f(z, ϕ) := (\frac{1}{c(zρ) − s(zρ) cos ϕ})^{2}

where ρ := dist(x, γx), c(zρ) := cosh zρ and s(zρ) := sinh zρ. Set Θ := Θ(x, γ^{−1}x, ξ).

Let A denote the normalized area measure on S_{x}. Giving S_{x} the spherical coordinates (θ, ϕ), we have dA = \frac{1}{4π} sin ϕdϕdθ.

Let ϕ′ := cos^{−1}(1 − 2α), and let C ⊂ S_{x} denote the spherical cap given by ϕ ≤ ϕ′, so that A(C) = α. Then by hypothesis (ii) we have ν(S_{∞}) ≤ A(C).

By Proposition 2.12, we have that Φ_{x}^{∗}𝔐_{x}^{2} ≤ A, which gives Φ_{x}^{∗}ν ≤ A by hypothesis (i). Set ν′ := Φ_{x}^{∗}ν and Θ′^{2} := Θ ∘ Φ_{x}. Then, by Proposition 2.19 (II) we have

∫_{S_{x}} Θ′^{2}dν′ ≤ ∫_{S_{x}} P_{1}^{2}(x, y, ϕ)dν′.

Note that by the definitions of P_{1}^{2} and C, we have inf_{C} P_{1}^{2} ≥ sup_{S_{x}−C} P_{1}^{2}. Hence

∫_{S_{x}} P_{1}^{2}dν′ = ∫_{C} P_{1}^{2}dν′ + ∫_{S_{x}−C} P_{1}^{2}dν′
≤ ∫_{C} P_{1}^{2}dν′ + ( sup_{S_{x}−C} P_{1}^{2}) ν′(S_{x} − C)
≤ ∫_{C} P_{1}^{2}dν′ + (inf_{C} P_{1}^{2}) A(C) ≤ ∫_{C} P_{1}^{2}dν′ + ∫_{C} P_{1}^{2}dA
≤ ∫_{C} P_{1}^{2}dA.

By the formula for P_{1}^{2}, we have P_{1}^{2} = f(1, ϕ). Therefore

∫_{C} P_{1}^{2}dA = ∫_{C} f(1, ϕ)dA = \frac{1}{4π} ∫_{0}^{2π} ∫_{0}^{ϕ′} f(1, ϕ) sin ϕdϕdθ
= \frac{α}{(c(ρ) − s(ρ))(c(ρ) − s(ρ) + 2α(ρ))}.

Using hypothesis (iii) and last equation above and equations for c(ρ), s(ρ), we can solve for ρ. This gives

ρ ≥ \frac{1}{2} log \frac{β(1 − α)}{α(1 − β)}

which is the desired result. q.e.d.

Lemma 3.3. Given x, y nonnegative numbers with x ≥ y and x + y ≤ 1, we have

(I)  \frac{1 − x}{y} ≥\frac{1 − p}{p}
(II)  \frac{(1 − x)(1 − y)}{xy} ≥ (\frac{1 − p}{p})^{2}

where p := \frac{1}{2}(x + y).

Proof. We will only prove (I). The proof of (II) is similar and is given in [1]. Let us write x = p + z and y = p − z for some nonnegative number z ∈ ℝ. Then we have (p − z)(1 − x) = y(1 − p − z), which implies p(1 − x) + y(p − 1) = z − 2zp. But we have p ≤ \frac{1}{2}, which gives us p(1 − x) + y(p − 1) ≥ 0. Hence the result follows. q.e.d.

3.2 Theorem 1.1 and 1.2

Proof of Theorems 1.1 and Theorem 1.2. Write S = {γ_{1}, ..., γ_{k}}. Let Ψ := S ∐ S^{−1}, and let x ∈ M̃ be an arbitrary point. Then by Theorem 2.18, we have a Γ−invariant conformal density measure [μ_{x}]^{D} on S_{∞} and a family of Borel measures [ν_{ψ}]_{ψ∈Ψ} on S_{∞} which satisfies (1)–(3) of Theorem 2.18.

Denote ν_{γ_{j}}(S_{∞}) and ν_{γ_{j}^{−1}}(S_{∞}) by, κ_{j} and ω_{j} respectively for each 1 ≤ j ≤ k. Without loss of generality, we can assume κ_{j} ≤ ω_{j}. Now, from Theorem 2.18 (1) and (2) and the assumption κ_{j} ≤ ω_{j}, we get 0 ≤ ω_{j} ≤ 1 and 0 ≤ κ_{j} ≤ 1/2. It follows from Theorem 2.18 (3) that ∫_{S_{∞}} Θ^{D}(x, γ_{j}^{−1}x, ξ)dν_{γ_{j}} = 1 − ω_{j}. Hence we can invoke Lemma 3.1 with α = κ_{j} and β = 1 − ω_{j}. This gives

ρ_{j} := dist(x, γ_{j}x) ≥ \frac{1}{D} log \frac{1 − ω_{j}}{κ_{j}} .

By using Lemma 3.3 (I) with x = ω_{j}, y = κ_{j}, we get

exp(Dρ_{j}) ≥ \frac{1 − p_{j}}{p_{j}} ,

which implies that

∑_{1}^{k} \frac{1}{1 + exp(Dρ_{j})} ≤ ∑_{1}^{k} p_{j} = \frac{1}{2} ∑_{1}^{k}(ω_{j} + κ_{j}) = \frac{1}{2}.

Now let us also assume the hypothesis of Theorem 1.2. Then by Proposition 2.3 and Proposition 2.10 and Corollary 2.13 we have [μ_{x}] = c𝔐_{x}^{D} for some constant c. Since both are normalized probability measures, we have c = 1. By D = 2 and Theorem 2.18 (2) we get ν_{γ_{j}} ≤ 𝔐_{x}^{2}. Hence we can invoke Lemma 3.2 with α = κ_{j} and β = 1 − ω_{j} to obtain

ρ_{j} ≥ \frac{1}{2} log \frac{(1 − ω_{j})(1 − κ_{j})}{κ_{j}ω_{j}}.

Then, by using Lemma 3.3 (II), we get

ρ_{j} ≥ log \frac{1 − p_{j}}{p_{j}}

and hence

\frac{1}{1 + exp ρ_{j}} ≤ \frac{1}{2}(ω_{j} + κ_{j}).

Therefore

∑_{j=1}^{k} \frac{1}{1 + exp ρ_{j}} ≤ \frac{1}{2} ∑_{j=1}^{k}(ω_{j} + κ_{j}) = \frac{1}{2}.

q.e.d.

Proof of Corollary 1.3. Suppose M is rank-1 locally symmetric manifold, then it follows from results of Sullivan, Bishop-Jones [3] and Fernández-Melián [6] that the Hausdorff dimension of the conical limit set Λ_{c}(Γ) is equal to D. Hence we have D ≤ 𝔇, which gives the desired result. q.e.d.

Remark 3.4. In the proof of Theorem 1.2, the condition that Γ is divergent is used only to conclude that Γ is ergodic with respect to μ_{x}. Hence we can replace the condition that Γ is divergent by the condition that Γ is ergodic with respect to μ_{x}.

Theorem 3.5. Suppose that M = M̃/Γ is a 3-manifold, that Γ is free and that 𝔐_{x}^{2}(Λ(Γ)) > 0. Suppose that D = 2 and Γ is ergodic with respect to μ_{x}. Then

∑_{γ∈S} \frac{1}{1 + exp dist(x, γx)} ≤ \frac{1}{2}.

Theorem 1.1 yields the following corollaries.

Corollary 3.6. Let Γ be free on a generating set S = {α_{1}, ⋯, α_{k}}. If Γ is convex-cocompact, then

∑_{1}^{k} \frac{1}{1 + exp(𝔇 dist(x, α_{j}x))} ≤ \frac{1}{2}

It is resonable to believe that the equality of D and the Hausdorff dimension of conical limit set of Γ should remain true for the case of variable (pinched negative) curvature. If this is the case, the hypothesis that Γ is convex-cocompact can be removed from Corollary 3.6.

Corollary 3.7. Let Γ be free on a generating set S = {α_{1}, ⋯, α_{k}}. There exists at least 1 ≤ i ≤ k − 1 distinct generators such that

dist(x, α_{j_{i}}x) ≥ \frac{1}{D} log(2i + 1)

for any x ∈ M̃.

Corollary 3.8. Let Γ be free on a generating set S = {α_{1}, ⋯, α_{k}}. Let x ∈ M̃, and let us assume elements of S is arranged so that the displacement function of x ∈ M̃ under α_{1} has least value. Let m be a nonnegative interger. Then each element of the generating set \{α_{1}^{m}α_{2}, α_{1}^{m+1}α_{2}, α_{3}, ⋯, α_{k}\} have displacement value ≥ \frac{log 3}{D}.

Proof. It is sufficient to prove this for Γ of rank 2. The general case follows from induction on k. Let {α_{1}, α_{2}} be any free basis of Γ, and denote the critical exponent of Γ by D_{Γ}. By Corollary 3.7 we have at most one generator say α_{1} such that dist(x, α_{1}x) < \frac{log 3}{D_{Γ}}. Then for any integer m ≥ 0, {α_{1}, α_{1}^{m}α_{2}} is also a free basis for Γ by Nielsen’s transformation, hence dist(x, α_{1}^{m}α_{2}x) ≥ \frac{log 3}{D_{Γ}} by Corollary 3.7. q.e.d.

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