LECTURE OUTLINE SEGOVIA LECTURES, JUNE 1996 CHRISTOPHER BISHOP

LECTURE 1: Introduction

(1) Pictures of limit sets.

(2) Möbius transformations; elliptic, parabolic, loxodromic.

(3) Definition of Kleinian group.

(4) Definition of limit set.

(5) Definition of conical limit set.

(6) Definition of ordinary set.

(7) Examples:

(a) Elementary

(b) Fuchsian

(c) Schottky groups

(8) Teichmüller space

(9) Degenerate groups

(10) Action on hyperbolic space.

(11) Quotient manifold

(12) Interpret limit set and conical limit set in terms of geodesic rays.

(13) Classes of groups:

(a) Finitely generated.

(b) Geometrically finite.

(c) Analytically finite.

(d) Topologically tame.

(14) Minkowski dimension.

(15) Whitney decomposition.

(16) Alternate definition of Minkowski dimension.

(17) Hausdorff content, general gauges.

(18) Hausdorff dimension.

(19) Hausdorff measure.

(20) Easy example where they differ.

(21) McMullen set.

(22) State some theorems:

(a) If Λ is connected then either it is a circle or it has dimension > 1. (Generalization of Bowen’s dichotomy; also related to Mostow rigidity.)

(b) If G is analytically finite then G is is geometrically finite iff dim(Λ) < 2.

(23) The Poincaré series.

(24) The critical exponent.

(25) The orbital counting function.

(26) Alternate formulation in terms of S^{2}.

(27) The hyperbolic metric on a plane domain.

(28) Uniformly perfect sets.

(29) Limit sets are uniformly perfect.

(30) Lemma: δ ≤ \overline{Mdim}(Λ). Equality if Ω/G is compact and Λ has zero area.

LECTURE 2: The Critical Exponent

(1) Review defn of dimension. Compute dimension of “circular” Cantor set.

(2) Review definitions of critical exponent.

(3) Definition of conical limit set.

(a) Non-tangential cones.

(b) Recurrent geodesics.

(4) Definition of bounded limit set Λ_{b}.

(5) Theorem: For any non-elementary Kleinian group, δ = dim(Λ_{c}) = dim(Λ_{b}).

(6) Historical remarks

(a) Results of Jarnik on dim of reals which are badly approximated by rationals.

(b) Patterson, Sullivan δ = dim(Λ_{c}) for Fuchsian and geometrically finite Kleinian groups.

(c) Fernandez and Melian proved δ = dim(Λ_{b}) for all Fuchsian groups and geometrically finite Kleinian groups.

(d) Stratmann similar but weaker in Fuchsian case.

(e) Also some results on S^{2} \ Λ_{c} and Λ \ Λ_{c} by Fernádez, LLorente, Melián, Nicolau. For example if R is a hyperbolic Riemann surface of infinite area without Greens function then escaping geodesics have dimension 1 (but zero Lebesgue measure since Greens function does not exist). Many interesting applications to holomorphic functions .

(f) Our proof works for rank 1 symmetric spaces.

(g) Paulin has shown that a slight variation works for the hyperbolic groups

of Gromov.

(7) Easy direction: dim(Λ_{c}) ≤ δ. Use obvious covering.

(8) Recall defn of dimension; it is enough to build Frostman type measure. Compute dimension of “circular” Cantor set.

(9) Lemma: Suppose E is formed by successively replacing disks D by a finite, disjoint subcollection {D_{j}} ⊂ D such that diam(D) ≤ Cdiam(D_{j}) for all j an d ∑_{j} diam(D_{j})^{α} ≥ diam(D_{j})^{α}. Then dim(E) ≥ α.

(10) Proof of Lemma: easy to construct measure. It has correct estimate for constructed disks. For other disk D let D_{0} be largest constructed disk which hits D but 2D_{0} does not contain D. Let D_{1} be parent of D_{0}. Then 2D_{1} contains D and is of comparable size (otherwise 2D_{0} would contain D).

(11) Hard direction: dim(Λ_{b}) ≥ δ.

(a) Idea of tree with vertices in orbit.

(i) If w ∈ C(z) then w ∈ B(z, N(1 − |z|)). (ii) If w ∈ C(z), then C_{0}^{−1} ≤ (1 − |w|)/(1 − |z|) ≤ 1/2. (iii) For w_{1}, w_{2} ∈ C(z) distinct, B(w_{1}, 2N(1 − |w_{1}|)) and B(w_{2}, 2N(1 −

|w_{2}|)) are disjoint. (iv) ∑_{w∈C(z)}(1 − |w|)^{δ−2ϵ} ≥ C_{0}^{2}(1 − |z|)^{δ−2ϵ},

(b) Choose point on sphere so Poincare series diverges locally.

(c) Choose scale r so every r-ball is moved off itself by some element. B_{1} = B(x, r).

(d) Let B_{2} be a disjoint image of B_{1}.

(e) Let θ be angular separation.

(f) Choose N so that: if z ≠ 0 then any geodesic ray starting at z and making angle at least θ/2 with the radial segment [0, z] stays inside the (Euclidean) ball B(z, N(1 − |z|)).

(g) Choose K so that: if the geodesic segment from z to w and making angle at least θ/2 with [0, z] and has hyperbolic length L then

1 ≤ \frac{(1 − |w|)e^{L}}{1 − |z|} ≤ K.

(h) Choose A so that: for w_{1}, w_{2} ∈ A_{n}, ρ(w_{1}, w_{2}) ≥ A implies

B(w_{1}, 3N(1 − |w_{1}|) ∩ B(w_{2}, 3N(1 − |w_{2}|)) = ∅.

(i) Now choose a annulus A_{n} so that the orbits V_{1} in have δ − ϵ sum very large. So large that thinning the points out to be distance A apart, the δ − ϵ sum is still large. Similarly for V_{2}.

(j) V_{1} and V_{2} are the first generation vertices. The children of v = g(0) ∈ V_{i} is either g(V_{1}) or g(V_{2}), depending on which makes a larger angle with the vertical. Continue this way to define whole tree.

(k) Verify the limit set of tree is in conical limit set.

(l) Verify that it is in the bounded limit set.

(12) Corollary: If Ω/G is compact and Λ has zero area then δ = dim(Λ_{b}) = dim(Λ_{c}) = dim(Λ) = \overline{Mdim}(Λ). In particular the Minkowski dimension exists.

(13) Result is still true if G is analytically finite. It is not true if we drop the zero area assumption. There are examples of analytically finite (even compact) groups where δ < 2 and Λ has positive area. (It could either be whole sphere or a nowhere dense subset).

(14) Corollary: δ(G) is lower semi-continuous with respect to algebraic convergence.

(15) Proof: If δ(G) > α, then there is a choice of a finite number of points in the proof above which gives a tree and a subset of dimension > α. For a small enough perturbation of the generators, this finite set does not move much so the α sum is still large enough to prove dim > α.

(16) Corollary: uniform estimates for the orbital counting function. Let {y_{j}} enumerate orbit of 0 and

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

Then N_{k} ≤ C_{ϵ}vol(B(0, 1)/G)^{−1}e^{(δ+ϵ)k}.

(17) Proof: If not then the proof of the theorem will let us build a tree whose boundary set has dimension > δ. The volume term arises when we have to “thin out” the orbits to make vertices more than distance A apart: a unit ball of hyperbolic space can contain at most Cvol(B(0, 1)/G)^{−1} orbit points, where C is an absolute constant.

LECTURE 3: Traveling Salesman and Bowen’s Dichotomy

(1) Theorem: If G is analytically finite and Λ is connected then either Λ is a circle or it has dimension > 1.

(2) Historical comments:

(a) This is Bowen’s theorem for co-compact quasi-Fuchsian groups.

(b) Sullivan proved it for geometrically finite groups.

(c) If G is a quasi-conformal deformation of a divergence type Fuchsian group one can show that Λ is either a circle or non-rectifiable in every arc.

(d) Astala and Zinsmeister showed this is false for every convergence group.

(e) Yue and others have extended Bowen’s result to higher dimensional Lie groups.

(f) This result was first of Bishop-Jones collaboration on Kleinian groups. Led to other results. Problem was raised to us by Ed Taylor via Canary-Taylor paper on limit sets with small dimension.

(3) First proof: using traveling salesman.

(4) Define β’s.

(5) State Jones traveling salesman theorem.

(6) State ϵ-close version.

(7) Define uniformly wiggly sets.

(8) Theorem: Connected, uniformly wiggly sets have dimension > 1.

(9) Show that if Ω/G is compact then Λ is either a circle or uniformly wiggly.

(10) Define singly and doubly cusped parabolics.

(11) Indicate problem with proof when there are doubly cusped parabolics.

(12) Second proof: using Schwarzian derivative.

(13) Consider a single simply connected component Ω_{0} of Ω(G).

(14) Let Φ be Riemann mapping onto Ω.

(15) State Plessner’s theorem.

(16) Define inner tangents, twist points.

(17) State McMillan’s twist point theorem.

(18) Define Fuchsian equivalent.

(19) Define Schwarzian derivative.

(20) State transformation laws. Deduce that |S(Φ)|(1 − |z|^{2})^{2} is constant on orbits of Ĝ.

(21) State relation to univalent functions.

(22) State Green’s formula.

(23) Deduce that Schwarzian integral is lower bound for length.

(24) Deduce that either Schwarzian is zero everywhere or integral is infinite. This is easiest if Ĝ is cocompact, but is still true as along as Ĝ is divergence type (so orbits are non-tangentially dense).

(25) If G is cocompact, the integral must grow exponentially fast.

(26) Easy to modify argument to work with parabolics.

(27) Deduce general case from component case.

(28) Circle packings.

(29) Larman’s theorem.

(30) Appolonian packing.

LECTURE 4: Brownian Motion, Manifolds and the Ahlfors conjecture

(1) Ahlfors Conjecture: If G is finitely generated then the limit set is either all of S^{2} or it has zero area.

(2) Historical remarks:

(a) Ahlfors proved this for geometrically finite groups.

(b) Bonahon, Thurston proved this for some geometrically infinite groups.

(c) Canary proved this for all topologically tame groups.

(d) Canary and Minsky have proven it for certain limits of topologically tame groups.

(e) It is known to be false for some analytically finite groups.

(f) Conjecture: topologically tame = finitely generated.

(3) Theorem: If G is analytically finite, geometrically infinite and δ < 2 then Λ has positive area.

(4) Corollary: IF G is analytically finite and geometrically infinite then dim(Λ) = 2.

(5) Independently Sullivan and Tukia has shown that geometrically finite groups have dimension < 2. If there are no rank 2 cusps then this is easy because then the limit set is porus.

(6) How to tell if limit set has positive area?

(7) Define convex hull of limit set.

(8) Does Brownian motion stay inside the convex hull forever? Equivalently, is there a non-zero bounded harmonic function which vanishes in the geometri-

cally finite ends?

(9) Recall definitions of geometrically infinite:

(a) No finite sided fundamental polyhedron.

(b) Uncountably many non-conical points.

(c) Convex core has infinite volume (more generally, unit neighborhood has infinite volume).

(d) Thick part of convex core is non-compact.

(10) Margulis lemma, description of thin parts.

(11) Definition of hyperbolic Laplacian.

(12) Definition of heat kernel. Interpretation in terms of Brownian motion.

(13) Estimates of decay of the heat kernel.

(14) The base eigenvalue; alternate definitions.

(15) Cheeger’s and Buser’s inequality

(16) Elstrodt-Patterson formula.

(17) General theory gives picture, but all the spectral theory can be avoided. (a) Heat kernel on ball.

(b) Heat kernel on quotient is given by summing. Can be estimated using orbital counting function.

(c) Orbital counting function has uniform bounds depending only on local volume.

(d) This gives heat kernel estimates on diagonal. (exponential decay divided by local volume).

(e) Semi-group property gives L^{2} estimate.

(f) Symmetry and Cauchy-Schwarz gives off diagonal estimate.

(18) Proof of Theorem:

(a) First fact: boundary of convex core has finite area. Sufficient to use approximating surface.

(b) Second fact: there is sequence of points in the thick part of the convex core whose distance to the boundary of the convex core tends to infinity.

(c) If Brownian motion is started at a thick point far enough inside the convex core, then with positive probability it never leaves the convex core.

LECTURE 5: The Law of the Iterated Logarithm for Kleinian Groups

(1) Recall definition of Hausdorff measure and dimension.

(2) Computing Hausdorff measure for a guage function is a more precise calculation than computing dimension.

(3) Historical remarks:

(a) Sullivan showed that for convex, compact groups the limit set has positive, finite δ-dimensional measure. He showed this was still true if there are only rank 1 cusps, but that if there are any rank 2 cusps then the δ-dimensional measure of the limit set is zero. Equivalently, if δ ≥ maximum of the parabolic ranks, then Λ has finite, positive δ-measure and in the other cases it has zero δ-measure.

(b) Tukia has refined this by showing that Λ has infinite measure for the guage | log t|^{ϵ}t^{δ} if ϵ(2δ − k_{max}) > 1.

(c) We know from earlier lectures that geometrically infinite limit sets have dimension 2.

(d) Sullivan created examples where the limit set had positive measure for

the gauge t^{2} log \frac{1}{t}.

(4) Theorem: Suppose G is analytically finite but geometrically infinite Kleinian group 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.

(5) Corollary: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.

(6) Compare this to a result of Fernandez and Melian for Fuchsian groups: if R has infinite area and no Green’s function then the escaping geodesics have dimension 1.

(7) Better estimates for the measure of the conical limit set are possible.

(8) Theorem: 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).

(9) Proof of Minkowski estimate:

(a) Shows that ∫_{M} k(x, y, t)dy ≃ t^{−1/2}.

(b) Use summing over orbits to show that there is at least on level where

the numb er of orbits is at most e^{2n}n^{−1/2}.

(10) Sullivan’s estimate (a) Construct U with at most linear on M

(b) List to disk.

(c) Write as Poisson integral of positive measure.

(d) Show measure satisfies μ(D(x, r)) ≤ Cr^{2} log \frac{1}{r}.

(11) This is best pointwise estimate possible. To improve the result we need to replace uniform estimate with an a.e. estimate.

(12) Frostman-Rogers-Taylor: Let μ be finite measure on [0, 1] and let A ⊂ [0, 1] be a closed set. Then

(a) If for all x ∈ A

lim sup_{n→∞} \frac{μ(I_{n}(x))}{φ(I_{n}(x))} > C,

then H^{φ}(A) ≤ H̃^{φ}(A) ≤ C^{−1}μ(A).

(b) If for all x ∈ A

lim sup_{n→∞} \frac{μ(I_{n}(x))}{φ(I_{n}(x))} < C,

then H̃^{φ}(A) ≥ C^{−1}μ(A).

(13) Proof of first part: Fix ϵ > 0 and choose an open set V ⊃ A with μ(V) close to μ(A). For every x ∈ A we can find n as large as we wish and such that

\frac{μ(I_{n}(x))}{φ(I_{n}(x))} > C.

Take n(x) to be the minimal integer satisfying this condition and such that b^{−n} ≤ ϵ and I_{n}(x) ⊂ V. Now {I_{n(x)}(x)} is a cover of A, and suppose {J_{k}} is a disjoint subcover. This covering of A has the property that that |J_{k}| ≤ ϵ for all k (by our choice of n(x)) and

∑_{k} φ(J_{k}) ≤ μ(V).

This implies

H̃_{ϵ}^{φ}(A) ≤ C^{−1}μ(A).

Therefore H̃^{φ}(A) ≤ C^{−1}μ(A).

(14) Proof of second part: Let

A_{m} = {x ∈ A : μ(I_{n}(x)) < C|I_{n}(x)| for all n > m}.

A trick of Frostman is to look at the set A_{m} instead of A. Since we have A = ∪_{m}A_{m} and A_{m+1} ⊃ A_{m} we have μ(A) = lim_{m→∞} μ(A_{m}). Thus it suffices to prove the estimate for A_{m}. Fix ϵ < b^{−m} and consider any cover of A by b-adic intervals {J_{k}} with |J_{k}| ≤ ϵ. With out loss of generality we assume every interval in the covering intersects A. Then

∑_{k} φ(J_{k}) ≥ ∑ φ(J_{k}) ≥ ∑ μ(J_{k}) ≥ C^{−1}μ(A_{m}).

This shows H̃_{(ϵ)}^{φ}(A) ≥ C^{−1}μ(A_{m}). Taking ϵ → 0 and m → ∞ gives the desired result.

(15) Law of the iterated Logarithm for harmonic functions: If U is harmonic and has a bounded gradient then

lim sup \frac{U(rx)}{\sqrt{log \frac{1}{1−r} log log log \frac{1}{1−r}}} < C < ∞,

for almost every x ∈ S^{2}.

(16) State LIL for random walks, Brownian motion.

(17) Given the LIL for harmonic functions prove that Λ has positive measure for t^{2}\sqrt{log 1t log log 1t}.

(18) To prove what we want we need to show the LIL holds for almost every radius with respect to μ measure (not just Lebesgue measure).

(19) To prove upper bound, we must find covering. Can’t be a covering by disks all of the same size. First cover by squares where U has exceeded LIL growth. Then by leftover squares.

(20) Sketch proof for random walk case.