GLOSSARY FOR KLEINIAN GROUPS SEGOVIA LECTURES, JUNE 1996 CHRISTOPHER BISHOP

Hyperbolic ball: We let 𝔹^{n} denote the unit ball of ℝ^{n} equipped with the hyperbolic metric

dρ = \frac{2|dz|}{1 − |z|^{2}} .

Hyperbolic half space: We will also use the upper half space model ℝ_{+}^{3} = {(x_{1}, x_{2}, y) : y > 0} for hyperbolic space with the metric

dρ = \frac{|dz|}{y}.

Hyperbolic volume: The hyperbolic volume in ℝ_{+}^{n} will be denoted

dV = \frac{dx_{1} ⋯ dx_{n−1}dy}{y^{3}} .

Möbius transformation: A Möbius transformation on the Riemann sphere is a mapping of the form

τ(z) = \frac{ax + b}{cz + d}.

These are also called linear fractional transformations. Such maps extend to be isometries of the hyperbolic metric on 𝔹. Möbius transformations may be identified with the group PSL(2, ℂ), since multiplying the top and bottom by the same scalar does not change the mapping.

Elliptic: The map τ is call elliptic if it is conjugate to a map of the form z → e^{it}z. In terms of the action on 𝔹, τ is elliptic if it has a fixed point in 𝔹.

Parabolic: τ is parabolic if it is conjugate to z → z + 1. In terms of the action on 𝔹, τ is parabolic if it has no fixed point

inf{ρ(z, τ(z)) : z ∈ 𝔹} = 0,

Loxodromic: τ is loxodromic if it is conjugate to z → λz with |λ| ≠ 1. This is equivalent to

inf{ρ(z, τ(z)) : z ∈ 𝔹} > 0.

Hyperbolic: τ is called hyperbolic if it is loxodromic and λ is real valued.

Trace: If τ(z) = \frac{ax+b}{cz+d} is normalized so |ad − bc| = 1, the trace of τ is a + d. This is not well defined on PSL(2, ℂ), but the square tr^{2}(τ) is. τ is elliptic iff tr^{2}(τ) ∈ [0, 4). τ is parabolic iff tr^{2}(τ) = 4. τ is hyperbolic iff tr^{2}(τ) ∈ (4, ∞). τ is strictly loxodromic iff tr^{2}(τ) ∉ [0, ∞).

Jørgensen’s inequality: If τ and σ generate a discrete group then

|tr(τστ^{−1}σ^{−1}) − 2| + |tr^{2}(τ) − 4 ≥ 1.

Selberg’s Lemma: If G is a finitely generated matrix group, then there is a finite index subgroup without torsion (i.e., no elements of finite order).

Kleinian Group: A discrete group G of isometries of the hyperbolic metric on 𝔹 is called a Kleinian group. In some references it is also required that there be a point of discontinuity on the sphere, i.e., a disk D on S^{2} which is disjoint from all its (non-identity) images under G.

Fuchsian group: A Kleinian group G is called Fuchsian if it fixes a disk in the Riemann sphere. Such a group is a group of isometries of the hyperbolic metric on the disk.

Extended Fuchsian group: G is an extended Fuchsian group if it has a index 2 Fuchsian subgroup, i.e., it is Fuchsian except there is also an element which switches the two components of Ω(G).

Elementary groups: G is called elementary if it contains a finite index Abelian subgroup.

Quasi-Fuchsian group: G is quasi-Fuchsian if it is conjugate to a Fuchsian group via a quasi-conformal homeomorphism of the sphere.

Totally degenerate: G is totally degenerate if Λ is connected and does not separate the plane.

Schottky group: G is a classical Schottky group if there is a collection of 2n disjoint disks {D_{i}} i = 0, …, 2n − 1, and G is generated by Möbius transformations g_{i} which map the outside of D_{2j} to the inside of D_{2j+1}, j = 0, …, n − 1. A general Schottky group is similarly defined, but the circles are replaced by closed Jordan curves.

B-group: A b-group is a finitely generated Kleinian group such that Ω(G) has a simply connected component.

Teichmüller space: Given a finite type surface S (compact with a finite number of punctures, possible none), the Teichmüller space T (S) is the set of equivalence classes of quasiconformal mappings of S to itself. Each such is represented by a Beltrami differential μ̃ which may be lifted to a Beltrami differential μ on the the upper halfplane, ℍ.

Bers embedding: Let Γ be a Fuchsian group acting on ℍ such that ℍ/Γ = S. There is a quasiconformal mapping F of the plane which fixes 0, 1, ∞ and such that \overline{∂}F/∂F = μ on ℍ and so that F is conformal on the lower half plane. On the lower half-plane the Schwarzian derivative S(F) satisfies

∥S(F)∥ = sup_{z} |Im(z)|^{2}|S(F)(z)| ≤ 6 < ∞.

This realizes T (S) as a bounded subset of a Banach space and gives a metric on T (S). This called the Bers embedding. The closure of T (S) with respect to this metric is denoted \overline{T (S)} and the boundary by ∂T (S). Points of \overline{T (S)} may be identified with certain Kleinian groups which are isomorphic to Γ. Moreover, convergence in the Teichmüler metric implies algebraic convergence of the groups.

Maskit embedding: This is an explicit description of Kleinian groups representing the Teichmüller space of a once puncture torus. For any μ ∈ ℂ let

T_{μ} = (\begin{matrix}−iμ & μ \\ −i & 0 \end{matrix}) , S_{1} = (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) .

Then the Teichmüller space can be identfied with a certain plane domain of μ values. Computer pictures of the domain have been drawn by Wright, but I do not know of a easy analytic description. e

Component subgroup: If Ω_{0} is a connected component of Ω, then G_{0} = {g ∈ G : g(Ω_{0}) = Ω_{0}} is the component subgroup corresponding to Ω.

Circle packing group: G is a circle packing group if Ω has infinitely many components and every one of them is a Euclidean disk. In other words, every component subgroup is Fuchsian.

Web group: G is a web group if every component subgroup is quasi-Fuchsian. Limit set: For a non-elementary G, the accumulation of any orbit in 𝔹 ∪ S^{2} is a

closed set Λ ⊂ S^{2} which is independent of the particular orbit. This is the limit set.

Ahlfors measure conjecture: If G is finitely generated then either Λ = S^{2} or it has zero area.

Conical limit set: The conical limit set Λ_{c} is the set of points x ∈ Λ for which there is a sequence of orbit points converging to x within a non-tangential cone in 𝔹. These are also called points of approximation. These correspond to geodesic rays in the quotient manifold which return to some compact set infinitely often.

Bounded limit set: The bounded limit set Λ_{b} is the set of points x ∈ Λ for which there is a sequence of orbit points {z_{n}} converging to x within a non-tangential cone in 𝔹 and such that sup_{n} ρ(z_{n}, z_{n+1}) < ∞. These correspond to geodesic rays in the quotient manifold which never leave some compact set.

Ordinary set: We call Ω = S^{2} \ Λ the ordinary set of G. G acts discontinuously on Ω, and Ω/G is a Riemann surface.

Area of Riemann surface: Suppose S is a genus g surface with P punctures and B branch points of order {n_{1}, … n_{B}}. The hyperbolic area of the surface is equal to

2π[2(g − 1) + P + ∑_{k=1}^{m}(1 − \frac{1}{n_{k}}).

Analytically finite: If Ω/G a Riemann surface of finite type (i.e., is a finite union of compact surfaces each with at most a finite number of punctures) we call G analytically finite.

Ahlfors finiteness theorem: This says says that any finitely generated Kleinian group is analytically finite. A refinement of Bers gives a bound for the hyperbolic area of Ω/G interms of the number of generators of G.

Geometrically finite: G is called geometrically finite if there is a finite sided fundamental polygon for G’s action on 𝔹. For Fuchsian groups this is the same as being finite generated, but there are finitely generated Kleinian groups which are not geometrically finite, i.e., which are geometrically infinite.

Geometric finiteness, 2nd version: G is geometrically finite iff every limit point of G is either a conical limit point or a fixed point of some parabolic element of G.

Quotient manifold: If G has no elliptics then M = 𝔹/G is a hyperbolic 3manifold. If G has elliptics then this quotient is an orbifold.

Convex core: If G is a Kleinian group with limit set Λ we let C(Λ) ⊂ 𝔹 denote the convex hull of Λ and let C(M) = C(Λ)/G ⊂ M = 𝔹/G be the convex core of M. C(M) can be defined more intrinsically as the closed convex hull of the closed geodesics in M.

Convex cocompact: G is called convex cocompact if C(M) is compact. This is the same as saying G is geometrically finite and has no parabolic elements. This is usually the easiest class of groups to deal with.

Geometric finiteness, 3rd version: G is geometrically finite iff the unit neighborhood of C(M) has finite hyperbolic volume. If G is analytically finite then G is geometrically finite iff C(M) itself has finite hyperbolic volume.

Injectivity radius: If x ∈ M then the injectivity radius inj(x) is half the length of the shortest loop through x; alternatively it is half the distance between the two closest distinct lifts of x to 𝔹. It is often convenient to assume that the injectivity radius of M is bounded away from 0, but this is a very special case. If G is topologically tame then there is always an upper bound for the injectivity radius inside the convex core.

Thin part: M_{thin}(ϵ) = {x ∈ M : inj(x) < ϵ}. The remainder of M is called the thick part.

Margulis Lemma: There is an ϵ > 0 (independent of G) so that if y ∈ 𝔹, then the group G_{ϵ}(x) generated by

{g ∈ G : ρ(x, g(x)) ≤ ϵ},

is almost-nilpotent and hence elementary.

If G has no elliptic elements, this says that there is an ϵ > 0 (independent of G) so that if

M_{thin(ϵ)} = {x ∈ M : inj(x) ≤ ϵ},

then every component of M_{thin(ϵ)} is one of four kinds:

(1) a torus cusp, i.e., a horoball in 𝔹 modulo a rank 2 parabolic subgroup.

(2) a rank one cusp, i.e., a horoball in 𝔹 modulo a rank 1 parabolic subgroup.

(3) a solid torus neighborhood of a closed geodesic, i.e., a r-neighborhood of a geodesic in 𝔹 modulo powers of a loxodromic element fixing the geodesic and translating it by some hyperbolic distance ≤ ϵ.

Moreover, the components of M_{thin(ϵ)} (i.e., the horoballs in (1) and (2) and the neighborhoods in (3)) may be taken to be pairwise disjoint.

Geometric finiteness, 4th version: Let

T_{ϵ}(G) = {x ∈ 𝔹 : G_{ϵ}(x) is infinite }.

G is geometrically finite iff C(M)\T_{ϵ}(G) is compact for some ϵ less than the Margulis constant. If G has no elliptics this is the same as requiring C(M) \ M_{thin}(ϵ) be compact.

Topologically tame: G is called topologically tame if M is homeomorphic to the interior of a compact 3-manifold with boundary. Then M can be written a the union of a compact part and a finite number of noncompact “ends” which are homeomorphic to S_{i} × ℝ_{+}, for some surface S_{i}.

Geometrically finite groups are topologically tame, which in turn are finitely generated.

Geometrically finite end: An end is geometrically finite if its lift to 𝔹 meets S^{2} in the ordinary set. Otherwise the end is geometrically infinite.

Analytically tame: M is analytically tame, i.e., there are finite area surfaces which sweep out the geometrically infinite ends. Canary proved that this is equivalent to topological tameness.

Critical exponent: The Poincaré series for G is

∑_{G} exp(−sρ(0, g(0))),

and the critical exponent is

δ(G) = inf{s : ∑_{G} exp(−sρ(0, g(0))) < ∞.

For any Kleinian group 0 < δ ≤ 2.

The critical exponent may also be defined using the series

∑_{G}(1 − |g(0)|)^{s},

or the series

δ_{Poin}(G) = inf{s : ∑_{g∈G} dist(g(z_{0}), Λ)^{s} < ∞},

where z_{0} ∈ Ω and distance is in the spherical metric.

Convergence group: G is said to be of convergernce type if

∑_{G}(1 − |g(0)|)^{2} < ∞.

Otherwise it is of divergence type.

Base Eigenvalue: For a complete Riemannian manifold M the number λ_{0} may be defined in numerous equivalent ways: : 1. The L^{2} spectrum is contained precisely in (−∞, λ_{0}].

2. λ_{0} is the infimum of the values λ for which there is a positive eigenfunction for Δ.

3. The infimum of ∫_{M} |∇u|^{2}/ ∫_{M} |u|^{2} over all smooth u with compact support. Cheeger’s inequality: The Cheeger constant of a 3-dimensional manifold M is

defined by

h(M) = inf \frac{area(∂(N)}{vol(N)},

where the inf is overall compact 3-dimensional submanifolds with Lipschitz boundary. Cheeger’s inequality says that

λ_{0}(M) ≥ \frac{1}{4}h(M)^{2}.

Buser’s inequality: This says that if the Ricci curvature of a 3-manifold M is bounded below by −2κ^{2}, then λ_{0}(M) ≤ Cκh(M). For hyperbolic manifolds the curvature is a negative constant.

Elstrodt-Patterson formula: If λ_{0} is the base eigenvalue of M then

λ_{0} = \begin{cases}1, & if δ < 1 \\ δ(2 − δ), & if δ ≥ 1. \end{cases}

The fact we will use is merely the special case that δ(G) < 2 implies λ_{0} > 0.

Orbital counting function: Given x, y ∈ M = 𝔹^{3}/G, assume x corresponds to an orbit containing 0 via the covering map and y to an orbit {y_{j}}. Let ρ = ρ(x, y) and

N_{k} = #({y_{j} : ρ + k ≤ ρ(0, y_{j}) < ρ + k + 1}).

Alternate critical exponents: We define

δ_{inrad}(K) = inf{s : ∑_{j} inrad(Ω_{j})^{s} < ∞},

δ_{diam}(K) = inf{s : ∑_{j} diam(Ω_{j})^{s} < ∞},

Here “inrad” denotes the radius of the largest disk in Ω_{j}. Distances are measured in the spherical metric, although if we normalize so that ∞ ∈ Ω and we ignore the single unbounded component we may use the Euclidean metric without changing the exponents.

Algebraic convergence: Suppose {G_{n}} is a sequence of m-generated Möbius groups each with a specific listing of its generators G_{n} = {g_{1n}, … g_{mn}}. We say that G_{n} converges algebraically to a Kleinian group G with generators {g_{1}, …, g_{m}} if g_{jn} → g_{j} for each 1 ≤ j ≤ m, as elements of PSL(2, ℂ). If we identify groups with points in PSL(2, ℂ)^{m}, this is just convergence in the product topology.

Polyhedral convergence: We say a sequence {G_{n}} converges polyhedrally to a group H if H is discrete and for some x_{0} ∈ 𝔹, the fundamental polyhedra (the Dirichlet polyhedron)

P (G_{n}) = {z ∈ 𝔹 : ρ(z, x_{0}) ≤ ρ(z, g(x_{0})) for all g ∈ G_{n}},

converge to P (H) uniformly on compact subsets of 𝔹.

Geometric convergence: Given a sequence of groups {G_{n}} we define

Env{G_{n}} = {g ∈ PSL(2, ℂ) : g = lim_{n} g_{n}, g_{n} ∈ G_{n}},

and we say G_{n} → H = Env{G_{n}} geometrically if for every subsequence {G_{n_{j}}}, Env{G_{n_{j}}} = Env{G_{n}}. Proposition 3.10 of Jorgensen-Marden (1990) says that G_{n} converges geometrically to H if and only if it converges polyhedrally to H.

Strong convergence: {G_{n}} converges strongly if it converges both algebraically and geometrically.

Semi-continuous: A real valued function f is lower semi-continuous if all sets of the form {x : f(x) > α} are open. Equivalently, iff x_{n} → x implies lim inf f(x_{n}) ≥ f(x).

Hyperbolic metric: For any plane domain Ω, there is a hyperbolic metric dρ. defined by the covering map from the unit disk. For general domains it satisfies

\frac{1 + o(1)}{d(z) log d(z^{)}−1} ≤ dρ ≤ 2\frac{|dx|}{d(z)}.

where d(z) = dist(z, ∂Ω) and for simply connected domains it satisfies

\frac{1}{2} \frac{|dx|}{d(z)} ≤ dρ ≤ 2\frac{|dx|}{d(z)}.

Koebe \frac{1}{4} Theorem: If f is univalent on a domain D and z_{0} ∈ D, then

\frac{1}{4}|f′(z_{0})|dist(z_{0}, ∂D) ≤ dist(f(z_{0}), ∂f(D)) ≤ 4|f′(z_{0})|dist(z_{0}, ∂D).

Uniformly perfect sets: A compact set K is called uniformly perfect if there is an ϵ > 0 such that for any x ∈ K and r < diam(K) there exists y ∈ K such that

ϵr ≤ |x − y| ≤ r.

There are several well known equivalent formulations of this condition. Suppose K is compact and Ω is its complement. Then the following are known to be equivalent:

(1) K is uniformly perfect.

(2) There is a positive lower bound for the length of the shortest closed hyperbolic geodesic in Ω.

(3) There is a constant C < ∞ so that

\frac{1}{C} \frac{|dz|}{d(z)} ≤ |dρ(z)| ≤ C \frac{|dz|}{d(z)}.

Conformal density: A conformal density of dimensional α is a positive measure μ on Λ with the property that

μ(g(E)) = ∫_{E} |g′|^{α}dμ, for all g ∈ G.

From this it follows that the Poisson extension of μ is a G-invariant harmonic function on 𝔹. There is a always a conformal density of dimension δ, where δ is the critical exponent of G (see Nicholls book).

Minkowski dimension: Given a compact set K let N(r, K) be the minimal number of disks of radius r needed to cover K then

\overline{dim_{M}}(K) = lim sup_{r→0} \frac{log N(r, K)}{log 1/r},

\underline{dim_{M}}(K) = lim inf_{r→0} \frac{log N(r, K)}{log 1/r},

are the upper and lower Minkowski dimension of K. If they agree the common value is the Minkowski dimension and is denoted dim_{M}(K). This is some times called the box counting dimension.

Hausdorff dimension: Suppose φ is a increasing continuous function from [0, ∞) to itself such that φ(0) = 0. We define the

H_{s}^{φ} = inf{∑ φ(r_{j}) : E ⊂ ∪_{j}D(x_{j}, r_{j}), r_{j} ≤ s}.

The Hausdorff content of E ⊂ ℝ^{2} is H_{∞}^{φ}. If φ(t) = t^{α} we denote H^{φ} by H^{α}. The Hausdorff dimension of E is

dim(E) = inf{α : H_{∞}^{α}(E) = 0},

and the Hausdorff measure of E is

H^{φ}(E) = lim_{s→0} H_{s}^{φ}(E),

Note that dim(E) ≤ \underline{dim_{M}}(E) ≤ \overline{dim_{M}}(E).

McMullen set: One trivial example where Minkowski and Hausdorff diemnsion disagree is E = \{0\} ∪ \{1, \frac{1}{2}, \frac{1}{3}, … \}, because dim_{H}(E) = 0, but dim_{M}(E) = 1/2. A more interesting example is the McMullen set: this is defined by dividing a square into a 2 by 3 array of rectangles, choosing one on the top row and two on the bottom row, and repeating this inside each chosen rectangle, obtaing at the nth stage a collection of 3^{n} rectangles of size 2^{−n} × 3^{−n}. This set has Minkowski dimension

1 + log_{3} \frac{3}{2} = 1.36907 …,

but Hausdorff dimension

log_{2}(1 + 2^{log^{3}}2) = 1.34968.

Packing dimension: Packing dimension is obtained from upper Minkowski dimension by setting

dim_{P} (K) = inf sup_{n} \overline{dim_{M}}(K_{n}),

where the inf is over all countable coverings of K by closed sets {K_{n}}. One can also define a packing measure analogous to Hausdorff measure and define packing dimension in this way, but it is more complicated. See Mattila’s book.

Dyadic square: A dyadic square is one of the form [j2^{−n}, (j +1)2^{−n}]×[k2^{−n}, (k + 1)2^{−n}]. In the definitions of Minkowski or Hausdorff dimension it it often convenient to consider coverings only by dyadic squares. The resulting content and measure are easily seen to be comparable to the usual definitions and the dimension is exactly the same.

Frostman’s lemma: If E supports a measure μ such that μ(D(x, r)) ≤ Cr^{α} for every disk D(x, r), then it is trivial to see that H^{α}(E) ≥ C^{−1}μ(E) > 0 so dim(E) ≥ α. Conversely, if E has positive H^{α} measure then there is a measure on E satisfying this estimate.

Frostman-Rogers-Taylor density theorem: A more sophisticated version of this is

Theorem Let μ be a finite measure on [0, 1]^{2} and let A ⊂ [0, 1]^{2} be a closed set. Given a point x let Q_{n}(x) denote the nth generation dyadic square containing x. Then

(1) If for all x ∈ A,

lim sup_{n→∞} \frac{μ(Q_{n}(x))}{φ(2^{−n})} > C,

then H^{φ}(A) ≤ MC^{−1}μ(A).

(2) If for all x ∈ A,

lim sup_{n→∞} \frac{μ(Q_{n}(x))}{φ(2^{−n})} < C,

then H^{φ}(A) ≥ M^{−1}C^{−1}μ(A).

Whitney squares: Whitney decomposition of a open set Ω ⊂ ℝ^{n} is a collection of cubes {Q_{j}} (disjoint except for boundaries) so that Ω = ∪_{j}Q_{j} and such that

diam(Q_{j}) ≃ dist(Q_{j}, ∂Ω).

The existence of a Whitney decomposition for any open set is a standard fact in real analysis. One can simply take a maximal collection of dyadic cubes in Ω such that diam(Q) ≤ dist(Q, ∂Ω). In ℝ^{2}, this gives a decomposition satisfying diam(Q) ≤ dist(Q_{j}, ∂Ω) ≤ 4diam(Q_{j}).

Whitney exponent: Given a compact set K let Ω = ℝ^{n} \ K and suppose {Q_{j}} is a Whitney decomposition of Ω. Define

δ_{Whit}(K) = inf{s : ∑_{Q_{j},diam(Q_{j})≤1} diam(Q_{j})^{s} < ∞}.

We need to omit the Whitney squares of large diameter, or the series would not converge. It is easy to check that the definition does not depend on the particular choice of Whitney decomposition. For any compact set dim_{M}(K) ≥ δ_{Whit}. If in addition K has zero area, dim_{M}(K) = δ_{Whit}.

Hausdorff distance: The Hausdorff distance between the sets in a metric space is

H(E, F) = max_{x∈E} dist(x, F) + max_{y∈F} dist(y, E).

The β’s: Given a set E in the plane and a square Q we define β(Q) as

β(Q) = t^{−1} inf_{L∈L} sup_{z∈E∩3Q} dist(z, L),

where L is the set of all lines L intersecting Q. We also define a related quantity

δ(Q) = diam(Q)^{−1} inf_{L∈L} H(E ∩ 3Q, L ∩ 3Q),

where H(E, F) denotes the Hausdorff distance between the set.

Jones’ Traveling Salesman theorem: If E is a set in the plane then the shortest curve Γ which contains E has length comparable (with universal constants) to

diam(E) + ∑_{Q} β(Q)^{2}diam(Q),

where the sum is over all dyadic squares in the plane. Similarly, the length of the shortest curve which passes within ϵ of every point of E has length comparable to

diam(E) + ∑_{diam(Q)≥ϵ} β(Q)^{2}diam(Q),

where the sum is over all dyadic squares bigger than size ϵ.

A similar result is true in higher dimensions.

Uniformly wiggly: A connected set with the property that there is a β_{0} > 0 such that β_{E}(Q) > β_{0} for every Q with \frac{1}{3}Q ∩ E ≠ ∅ is called uniformly wiggly.

Theorem Suppose E ⊂ ℝ^{2} is a closed, connected set and is uniformly wiggly with constant β_{0}. Then dim(E) ≥ 1 + Cβ_{0}^{2}, where C is an absolute constant.

Schwarzian derivative: The Schwarzian derivative of a locally univalent function f is defined by

S(f)(z) = [\frac{f′′(z)}{f′(z)}]′ − \frac{1}{2}[\frac{f′′(z)}{f′(z)}]^{2}
= [\frac{f′′′(z)}{f′(z)}] − \frac{3}{2}[\frac{f′′(z)}{f′(z)}]^{2}.

If we write f′ = e^{φ} then it can be rewritten as

S(f)(z) = φ′′ − \frac{1}{2}(φ′)^{2}.

Composition laws: S(f) ≡ 0 iff f is a Möbius transformation and that S satisfies the composition law

S(f ∘ g) = S(f)(g′)^{2} + S(g).

In particular, if g is Möbius then

S(f ∘ g) = S(f)(g′)^{2} and S(g ∘ f) = S(f).

Schwarzian’s and univalent functions: Suppose f is analytic on 𝔻. If f is univalent and φ = log f′ then

|S(f)(z)| ≤ \frac{6}{(1 − |z|^{2})^{2}} and |φ′(z)| ≤ \frac{6}{1 − |z|^{2}} .

Conversely, if

|S(f)(z)| ≤ \frac{2}{(1 − |z|^{2})^{2}} or |φ′(z)| ≤ \frac{2}{(1 − |z|^{2})},

then f is univalent. If the 2 is replaced by a small constant, ϵ, then the image of the disk is a quasidisk with small constant.

Bloch space: The Bloch space consists of those analytic functions on the disk such that

sup_{z} \frac{|f′(z)|}{1 − |z|^{2}} < ∞.

Note that these are exactly the Lipschitz functions from the disk with the hyperbolic metric to the plane with the Euclidean metric.

Fuchsian equialent: If G is a Kleinian group and Ω_{0} is a simply connected component of the ordinary set let Φ : 𝔻 → Ω be a Riemann map. Then Ĝ = Φ^{−1}∘G∘Φ is a Fuchsian group and is called the Fuchsian equivalent of G.

Greens’ theorem for Lipschitz domains: Suppose 0 ∈ D and D is Lipschitz with constant M, that we have dist(0, ∂D) ∼ diam(D) = 1 and that F is holomorphic on 𝔻 and satisfies

∫∫_{D} |F′(z)|^{2}d(z)dxdy < ∞,

with d(z) = dist(z, ∂D). Then F ∈ H^{2}(D) (the Hardy space) and

∫_{∂D} |F (z)|^{2}ds(z) ∼ |F (0)|^{2} + ∫∫_{D} |F′(z)|^{2}d(z)dxdy

with constants depending only on M. Similarly,

∫_{∂D} |F (z)|^{2}ds(z) ∼ |F (0)|^{2} + |F′(0)|^{2} + ∫∫_{D} |F′′(z)|^{2}d(z)^{3}dxdy.

Non-tangential limits: We say that a function f on the disk has non-tangential limits at a point x on the boundary if f has a limit within any non-tangential cone with vertex x.

Non-tangentially dense: A set {z_{n}} ⊂ 𝔻 is called non-tangentially dense if almost every point of ∂𝔻 is a non-tangential limit point of the set. For orbits of Fuchsian groups this occurs iff the group is divergence type, i.e.,

∑_{g∈G} 1 − |g(0)| = ∞.

In particular, this happens whenever the Fuchsian group is finitely generated and of the first kind (i.e., has limit set the whole circle.)

Plessner’s theorem: For an analytic function F, Plessner’s theorem says that the unit circle 𝕋 can be divided into two sets E_{0}, E_{1} such that at almost every (Lebesgue measure) point of E_{0}, F has a finite, non-zero non-tangential limit and almost everywhere on on E_{1}, F is non-tangentially dense in the plane (i.e., for any complex value w there is a sequence {z_{n}} converging in a cone to x so that f(z_{n}) → w).

Inner tangent point: A point x ∈ ∂Ω is called a (inner) tangent point of Ω if for any θ < π, x is the vertex of a cone in Ω with angle θ, but not for any θ > π. The set of inner tangents is the same, up to a set of 1-dimensional measure zero, as the set E ⊂ ∂Ω of points which are vertices of some cone in Ω(G).

Twist points: If Ω is simply connected and x ∈ ∂Ω, then x is called a twist point of Ω if

lim sup_{z→x,z∈Ω} arg(x − z) = +∞, lim inf_{z→x,z∈Ω} arg(x − z) = −∞.

McMillan’s theorem: Suppose Ω is simply connected and Φ : 𝔻 → Ω is a Riemann mapping. Let ω denote the harmonic measure on Ω, i.e., ω(E) = |Φ^{−1}(E)|/2π. Then almost every point of ∂Ω is either the vertex of a cone in Ω or is a twist point and these two sets correspond almost everywhere (via Φ) to the sets E_{0} and E_{1} in Plessner’s theorem applied to Φ′, the derivative of the Riemann mapping Φ : 𝔻 → Ω.

The following are equivalent for a simply connected domain Ω:

(1) The set of inner tangents of Ω has zero 1-dimensional measure.

(2) The twist points have full harmonic measure.

(3) Harmonic measure is singular to 1-dimensional measure.

Vitali covering lemma: If E is a set and {I_{j}} is a collection of intervals such that each point of E is contained in intervals in {I_{j}} of arbitrarily small length, then there is a disjoint subcollection which covers almost every point of E.

Larman’s theorem: If {D_{j}}_{1}^{∞} is a collection of two or more disjoint open disks and E = D_{0} \ ∪_{j}D_{j} then dim(E) > 1 with an estimate independent of the collection.

Hyperbolic Laplacian: The hyperbolic Laplacian on 𝔹^{n} is given by

Δ_{H} = \frac{(1 − r^{2})^{2}}{4} [Δ + \frac{2(n − 2)r}{1 − r^{2}} \frac{∂}{∂r}],

and on the upper half space it is

Δ_{H} = y^{2}[Δ − \frac{n − 2}{y} \frac{∂}{∂y}].

Greens theorem: Greens theorem states that

∫∫_{Ω} GΔF − F ΔGdV = ∫_{∂Ω} G\frac{∂F}{∂n} − F \frac{∂G}{∂n}dA.

Poisson kernel: The hyperbolic Poisson kernel on 𝔹^{n} is

P (z, ζ) = w_{n}[\frac{1 − |z|^{2}}{|z − ζ|^{2}} ]^{n−1},

where z ∈ 𝔹, ζ ∈ S^{2} and w_{n} is chosen to make the integral 1. On the upper half space the hyperbolic Poisson kernel is

P (z, ζ) = v_{n}[\frac{y}{|z − ζ|^{2}} ]^{n−1}.

By way of comparison, the usual Poisson kernel on the ball is

P (z, ζ) = ω_{n} \frac{1 − |z|^{2}}{|z − ζ|^{n}} ,

and on the upper half-space is

P (z, ζ) = ν_{n} \frac{y}{(|ζ|^{2} + y^{2})^{\frac{n+1}{2}}} .

Heat kernel: The heat kernel k(x, y, t) on a manifold M is the function such that

u(x, t) = ∫_{M} f(y)k(x, y, t)dy,

is a solution of the heat equation u_{t} = Δu with initial value f. Equivalently, k(x, y, t)dt is the distribution of a Brownian motion on M started at x, at time t.

The heat kernel on 𝔹 is

k(w, z, t) = (4πt)^{−3/2} \frac{ρ(w, z)}{sinh(ρ(w, z))}e^{−t− \frac{ρ(w,z)^{2}}{4t}} .

Using the observation that

\frac{r}{sinh(r)} ≃ (1 + r)e^{−r},

for r > 0 gives,

k(w, z, t) ≃ t^{−3/2}(1 + ρ(w, z)) exp(−t − ρ(w, z) − \frac{ρ(w, z)^{2}}{4t}).

The heat kernel is symmetric, i.e., K(x, y, t) = K(y, x, t). Fix some n and recall that L^{2}(Ω_{n}) has a complete orthonormal basis of eigenfunctions for the Laplacian, {φ_{j}^{n}}_{j=0}^{∞}, with eigenvalues {λ_{j}^{n}} chosen so that λ_{j}^{n} ≤ λ_{j+1}^{n} Then

K_{n}(x, y, t) = ∑_{j=0}^{∞} φ_{j}^{n}(x)φ_{j}^{n}(y)e^{−λ_{j}^{n}t}

Green’s function: The Green’s function for a manifold is given by

G(x, y) = ∫_{0}^{∞} k(x, y, t)dt.

For a fixed y this is harmonic in x except for a pole at y. In terms of probabilty theory, ∫_{E} G(x, y)dy is the expected amount of time that a Brownian motion started at x will spend in the set E.

Base Eigenvalue: For a complete Riemannian manifold M the number λ_{0} may be defined in numerous equivalent ways: : 1. The L^{2} spectrum is contained precisely in (−∞, λ_{0}].

2. λ_{0} is the infimum of the values λ for which there is a positive eigenfunction for Δ.

3. The infimum of ∫_{M} |∇u|^{2}/ ∫_{M} |u|^{2} over all smooth u with compact support. Cheeger’s inequality: The Cheeger constant of a 3-dimensional manifold M is

defined by

h(M) = inf \frac{area(∂(N)}{vol(N)},

where the inf is overall compact 3-dimensional submanifolds with Lipschitz boundary. Cheeger’s inequality says that

λ_{0}(M) ≥ \frac{1}{4}h(M)^{2}.

Buser’s inequality: This says that if the Ricci curvature of a 3-manifold M is bounded below by −2κ^{2}, then λ_{0}(M) ≤ Cκh(M). For hyperbolic manifolds the curvature is a negative constant.

Elstrodt-Patterson formula: If λ_{0} is the base eigenvalue of M then

λ_{0} = \begin{cases}1, & if δ < 1 \\ δ(2 − δ), & if δ ≥ 1. \end{cases}

The fact we will use is merely the special case that δ(G) < 2 implies λ_{0} > 0.

BMO: The space BMO is defined to be the class of locally integrable functions f so that

∥f∥_{BMO} = sup_{I} – ∫_{I}|f(x) − – ∫_{I}f(y)dy|dx < ∞.

On ℝ this space contains L^{∞} and is contained in every L^{p} p < ∞. One of the most important estimates concerning

John-Nirenberg Theorem: If f ∈ BMO then

\{x ∈ Q : |f(x) − – ∫_{Q}f| > λ\}| ≤ C_{1}|Q| exp(−C_{2} \frac{λ}{∥f∥_{BMO}} ).

Carleson cube: A Carleson cube Q in ℝ_{+}^{n+1} is any cube with sides parallel to the coordinate axes and a face on ℝ^{n}.

Carleson measure: A measure μ on ℝ_{+}^{n+1} is called Carleson measure if there is a M < ∞ so that μ(Q) ≤ M|Q|_{n}, for all Carleson cubes Q. The best M is called the Carleson norm of μ and is denoted ∥μ∥_{C}.

Suppose u is the harmonic extension of f from ℝ^{n} to the upper half-space ℝ_{+}^{n+1}. Then f ∈ BMO iff

μ = |∇u|^{2}ydx_{1} … dx_{n}dy,

is a Carleson measure. Moreover, ∥f∥_{BMO} ≃ \sqrt{∥μ∥_{C}}.

Square Function: For x ∈ S^{2} let Γ_{α}(x) be the non-tangential cone with vertex x and angle α. Define the square function

Sf(x) = (∫_{Γ(x)} |∇f(y, t)|^{2}r^{−1}dydr)^{−1/2}.

Then a special case of a result of Chang, Wilson and Wolff is

Theorem: Suppose Sf ∈ L^{∞}(S^{2}). Then there are constants C_{1} > 0 and C_{2} < ∞ so that

∫_{S^{2}} exp(C_{1} \frac{|f − f(0)|^{2}}{∥Sf∥_{∞}^{2}} ) < C_{2}.

If u is a harmonic function with gradient |∇u| ≤ y^{−1}, then the Carleson measure characterization of BMO says that its restriction f to {y = 2^{−n}} is in BMO with norm at most ≃ \sqrt{n}. Thus the John-Nirenberg theorem says that ∫ e^{C|f|} < ∞ for some C > 0. The Chang-Wilson-Wolff theorem says something stronger; that ∫ e^{Cf^{2}} < ∞.

In fact we will use the following corollary of their result. For a function u defined on the ball, define the radial maximal function

RU(rx) = sup_{s<r} U(sx).

Corollary: Suppose Sf ∈ L^{∞}(S^{2}) and let u be its harmonic extension to the ball. Then there are constants C_{1} > 0 and C_{2} < ∞ so that

|\{x ∈ Q : |Rf(x) − – ∫_{Q}f| > λ\}| ≤ C_{1}|Q| exp(−C_{2} \frac{λ}{∥Sf∥_{∞}^{2}} ).

Dyadic martingale: Let Q_{0} be a unit dyadic cube. Let G_{n} be the σ-algebra geberated by dyadic cubes in Q_{0} of side length 2^{−n}.. Let E(f, G_{n}) be the conditional expectation of f on G_{n}, i.e.,

E(f, G_{n})(x) = \frac{1}{|Q_{n}(x)|} ∫_{Q_{n}(x)} f,

where Q_{n}(x) is the dyadic cube of size 2^{−n} containing x. A dyadic martingale is a seqeunce of functions {f_{n}} so that f_{n} is G_{n} measureable (i.e., is constant on each nth generation cube) and E(f_{n+1}, G_{n}) = f_{n}.

Martingale difference function: Let s_{n} = f_{n+1} − f_{n} and let

Sf(x) = ∑ n > 0∥χ_{Q_{n}(x)}s_{n}(x)∥_{∞}^{2}.

Theorem: Let {f_{n}} be a dyadic martingale on Q_{0} with limit function f. Then for λ ≥ 0,

|\{x ∈ Q_{0} : f(x) − f_{0}(x) ≥ λ\}| ≤ exp(−\frac{1}{2}λ^{2}/∥Sf∥_{∞}^{2}).

RESULTS

Theorem 1. If G is a non-elementary, discrete Möbius group on 𝔹 then δ(G) = dim(Λ_{c}(G)).

Corollary 2. Suppose G is discrete and has more than one limit point. If {G_{n}} is a sequence of discrete Möbius groups converging algebraically to G, then

δ(G) ≤ lim inf_{n→∞} δ(G_{n}).

Corollary 3. If G is any non-elementary, discrete Möbius group and ϵ > 0 then there is a R = R(ϵ) < ∞ such that given any point x ∈ M = 𝔹/G the set of directions (i.e., unit tangents at x) which correspond to geodesic rays starting at x which never leave the ball of radius R around x has dimension ≥ δ(G) − ϵ = dim(Λ_{c}(G)) − ϵ. In particular, the set of directions at x of bounded geodesic rays has dimension exactly δ(G).

Theorem 4. Suppose G is a finitely generated Kleinian group and Ω is a simply connected, invariant component of Ω(G). Then dim(∂Ω) = 1 iff ∂Ω is a circle.

Corollary 5. If G is a finitely generated Kleinian group then its limit set is either totally disconnected, a circle or has Hausdorff dimension > 1.

Theorem 6. If G is an analytically finite Kleinian group which is geometrically infinite then dim(Λ(G)) = 2.

Corollary 7. An analytically finite Kleinian group is geometrically finite if and only if dim(Λ(G)) < 2.

Corollary 8. Suppose that G is a non-elementary, analytically finite Kleinian group and that area(Λ(G)) = 0. Then δ(G) = dim(Λ).

Corollary 9. If G is an analytically finite Kleinian group then the Minkowski dimension of Λ exists and equals the Hausdorff dimension.

Corollary 10. If {G_{n}} is a sequence of N-generated Kleinian groups which converges algebraically to G then dim(Λ(G)) ≤ lim inf_{n} dim(Λ(G_{n})).

Theorem 11. Suppose G is analytically finite but geometrically infinite and the injectivity radius of M = 𝔹/G is bounded away from zero. Then Λ \ Λ_{c} has positive Hausdorff measure with respect to the gauge function φ(t) = t^{2}\sqrt{log \frac{1}{t} log log log \frac{1}{t}}. If, in addition, G is topologically tame and Λ ≠ S^{2}, then Λ has finite Hausdorff measure with respect to φ and Λ_{c} has zero measure for this gauge function.

Theorem 12. Let E = {x ∈ [0, 1] : S_{n}(x) > 0 for all n}. Then 0 < H^{φ}(E) < ∞ for φ(t) = t\sqrt{log \frac{1}{t} log log log \frac{1}{t}}.

Corollary 13. Suppose G is an analytically finite Kleinian group such that the injectivity radius of M = 𝔹/G is bounded away from zero. Then Λ \ Λ_{c} is either empty or has dimension 2, depending on whether G is geometrically finite or infinite.

Corollary 14. Suppose G is topologically tame, geometrically infinite, has a positive lower bound for the injectivity radius, and Λ ≠ S^{2}. Then μ = H^{φ} is a conformal density of dimension 2 where H^{φ} is the Hausdorff measure associated to the guage function φ(t) = t^{2}\sqrt{log \frac{1}{t} log log log \frac{1}{t}}

Corollary 15. Suppose G is a topologically tame Kleinian group and the injectivity radius of M = 𝔹/G is bounded away from zero. Then Λ_{c} has either full Lebesgue measure or zero Hausdorff measure with respect to φ(t) = t^{2}\sqrt{log \frac{1}{t} log log log \frac{1}{t}}, depending on whether Λ = S^{2} or not.

Theorem 16. If G is an analytically finite but geometrically infinite Kleinian group with the injectivity radius of M = 𝔹/G bounded away from zero, then N(Λ, t) ≥ C_{1}/ψ(t) where

ψ(t) = t^{2}\sqrt{log \frac{1}{t}}.

If, in addition, G is topologically tame and Λ ≠ S^{2}, then N(Λ, t) ≤ C_{2}/ψ(t).

Theorem 17. If G is an analytically finite Kleinian group then

∑_{j} diam(Ω_{j})^{2} < ∞,

where diameter is taken in the spherical metric.

Theorem 18. Suppose G is an analytically finite Kleinian group such that Ω(G) = S^{2} \ Λ(G) has infinitely many components. Then

δ_{Poin} = δ_{inrad} = δ_{diam}.

If, in addition, area(Λ) = 0 then they are all equal to dim_{H}(Λ).

Theorem 19. If G is a non-elementary, analytically finite Kleinian group with Λ ≠ S^{2}, then δ_{Poin}(G) = δ_{Whit}(Λ).

TECHNICAL LEMMAS

Lemma 20. If G is a Kleinian group, 0 is the center of 𝔹 and z ∈ Ω(G) ⊂ ∂𝔹, then d(g(z)) ∼ 1 − |g(0)|, for all g ∈ G with constants that depend on z and G, but not on g (here d means the spherical distance).

Lemma 21. If G is an analytically finite, non-elementary Kleinian group then Λ is uniformly perfect. In particular, if ρ is the hyperbolic metric on a component Ω of Ω(G), then

|dρ(z)| ∼ \frac{|dz|}{d(z)}.

Lemma 22. Suppose G is an analytically finite Kleinian group and M = 𝔹/G. Let {Ω_{j}}_{1}^{N} be the conjugacy classes of components of Ω(G) (i.e., the geometrically finite ends of M). For each Ω_{j} there is a locally Lipschitz surface S_{j} in M so that the following holds.

(1) dist(S_{j}, C(M)) > 2.

(2) Each S_{j} has hyperbolic finite area.

(3) The function f(y) = (vol(B(y, 1))^{−1/2} is integrable over S_{j}, i.e.,

∫_{S_{j}}(vol(B(y, 1))^{−1/2}dA(y) < ∞.

(B(y, 1) is the ball in M of hyperbolic radius 1 around y, vol denotes hyperbolic volume in M and dA is hyperbolic area on S_{j}).

(4) The surfaces S_{j} separate C(M) from the geometrically finite ends of M, i.e., there is an ϵ_{0} > 0 so that w(z) ≤ 1 − ϵ_{0} on the component M_{1} of M \ ∪_{j}S_{j} containing C(M).

(5) There is a constant C < ∞ so that the Hausdorff distance between S and ∂C(M) is less than C. In other words

S ⊂ {x ∈ M : dist(x, ∂C(M)) < C},

and

∂C(M) ⊂ {x ∈ M : dist(x, S) < C}.

Lemma 23. Suppose G is analytically finite and geometrically infinite. Then there is a sequence {x_{n}} ∈ C(M) with dist(x_{n}, ∂C(M)) → ∞ and inj(x_{n}) > ϵ for all n.

Theorem 24. If δ > 0 then there is a constant C = C(δ) such that

0 ≤ K(x, y, t) ≤ Cvol(B(x, t^{1/2}))^{−1/2}vol(B(y, t^{1/2}))^{−1/2}e^{−ρ(x,y)^{2}(4+δ)^{−1}t^{−1}},

for 0 < t < 1, where ρ(x, y) denotes the hyperbolic distance between x and y in M, and vol(B(x, t)) denotes the volume (in M) of the ball of radius t around x. Furthermore,

0 ≤ K(x, y, t) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{(δ−λ_{0})t}e^{−ρ(x,y)^{2}(4+δ)^{−1}t^{−1}},

for 1 ≤ t < ∞, where λ_{0} is the base eigenvalue for the Laplacian on M.

Lemma 25. Let G be any discrete Möbius group and let M = 𝔹/G. Then there is an absolute C < ∞ so that for any x, y ∈ M and t > 0 we have

K(x, y, t) ≤ \frac{C(1 + ρ(x, y))t^{−3/2}e^{−t}e^{− \frac{1}{8} t^{−1}ρ(x,y)^{2}}}{vol(B(x, 1))^{1/2}vol(B(y, 1))^{1/2}} ,

for 0 < t ≤ \frac{1}{8}ρ(x, y), and

K(x, y, t) ≤ \frac{Ce^{−λ_{0}t}}{vol(B(x, 1))^{1/2}vol(B(y, 1))^{1/2}} ,

for t ≥ 1. Here λ_{0} denotes the base eigenvalue of M.

Corollary 26. Let G be any discrete Möbius group and let M = 𝔹/G. Suppose the base eigenvalue λ_{0} is non-zero. Then there are constants C < ∞ and c > 0 (depending only on λ_{0}) so that for any x, y with ρ(x, y) ≥ 8, we have

G(x, y) ≤ Cvol(B(x, 1))^{−1/2}vol(B(y, 1))^{−1/2}e^{−cρ(x,y)}.

Lemma 27. Suppose M = 𝔹/G has injectivity radius bounded below by ϵ > 0 and that Greens function G(w, z) exists on M. Then

sup_{z∈M} sup_{ρ(w,z)≥1} |∇_{w}G(w, z)| ≤ A(ϵ) < ∞,

i.e., the gradient of Green’s function is uniformly bounded outside a neighborhood of the pole.

Lemma 28. Suppose M = 𝔹/G has injectivity radius bounded below by ϵ > 0 and that Greens function G(w, z) exists on M. Then there exists a positive harmonic function U on M such that

(1) sup_{z∈M} |∇U(z)| ≤ 1,

(2) U tends to zero in the geometrically finite ends of M. If, in addition, G is topologically tame then there are constants a_{1}, a_{2} and a_{3} so that

∫_{B(z,a_{1})} |∇U|^{2}dV ≥ a_{2},

for every z such that U(z) ≥ a_{3}.