REFERENCE LIST SEGOVIA LECTURES, JUNE 1996 CHRISTOPHER BISHOP

Background on hyperbolic geometry and Kleinian groups:

[15] Nice introduction to hyperbolic geometry.

[13] Discrete groups, mainly Fuchsian groups and properties to fundamental domains. Great reference for hyperbolic trigonometry.

[48] Introduction to heat equation.

[81] The book on Kleinian groups and combination theorems.

[90] The best reference for ergodic theory on limit sets. Carefully presents much of Sullivan’s work.

Background on analysis:

[96] Conformal mapping and harmonic measure.

[53] Univalent functions.

[101] Real analysis, maximal theorems, differentiation.

[83] Hausdorff and other dimensions. Rectifiable sets.

[56] Hausdorff dimension and measures.

Base eigenvalue:

[37] Estimates λ_{0} in terms of volume and boundary area of convex core.

[66] Heat equation on general Riemannian manifolds.

[50] Finite volume case.

[52] Good paper. Shows δ for classical Schottky groups is bounded away from 2.

[100] A lower bound for the first eigenvalue of a negatively curved manifold.

[109] Many characterizations of the base eigenvalue, including the relation to the Poincaré exponent.

Brownian motion:

[28] Shows dimension of outer boundary of Brownian motion is > 1.

[51] Conditioned Brownian motion.

[54] Applications of Brownian motion to analysis.

[72] Construction and basic properties of Brownian motion.

Circle packings:

[33] Technical results on circle packings.

[36] Great pictures of limit sets. Constructs examples of circle packing groups.

[71] Classification of certain circle packing groups.

[76] Non-trivial circle packings have dimension bounded uniformly away from 1.

[92] Shows Poincaré exponent agrees with dimension for circle packing groups. Extended by [26] and [21].

Classification of Kleinian groups:

[1] Example of limit sets points which are not on the boundary of any complementary component.

[2] Geometrically finite B-groups are on the boundary of Teichmüller space.

[4] Finitely generated groups can be built by combination theorems out of elementary groups and web groups.

[7] Structure of finitely generated Kleinian groups.

[73] Quasiconformal deformations of groups. Contains a list of all elementary groups.

[78] Maskit’s complement to [17].

[79] Combination theorems.

[80] Complementary components of limit sets have diameters tending to zero. Construction of certain geometrically infinite groups.

Conical limit set:

[14] Characterizes geometrically finite groups in terms of conical limit set. Extended by [22].

[95] Applies argument of [26] to show δ is dimension of conical limit set in the hyperbolic groups of Gromov.

[65] Gromov hyperbolic groups, by Gromov.

[121] Tukia’s paper on conical limit points and Myrberg points.

Convergence of Kleinian groups:

[70] Algebraic and geometric convergence of Kleinian groups.

Convex core:

[55] Details on work of Sullivan. Shows boundary of convex core is finite area Lipschitz surface.

[110] The volume of convex cores under algebraic and geometric convergence, Ed Taylor’s thesis.

Degenerate groups:

[17] Show degenerate groups are generic on boundary of Teichmüller space.

[62] Shows degenerate groups are geometrically infinite.

[85] Complements Bers paper by showing geometrically finite groups are dense on the boundary of Teichmüller space.

Dimension of limit sets:

[26] Shows δ is dimension of conical limit set. Proves geometrically infinite limit sets have dimension 2.

[27] Shows that uniformly wiggly sets have dimension > 1. Shows connected limit sets are either circles or have dimension > 1 (i.e., generalizes Bowen’s theorem, see next paper).

[32] A classic. Shows that certain quasiFuchisan limit sets are either circles or have dimension > 1

[41] Shows that if dimension of limit set if < 1 then group is geometrically finite. Classifies groups where dimension ≤=, except for one step filled by [27].

[103] Minkowski and Hausdorff dimension agree for geometrically finite groups. Done for all finitely generated groups in [26].

[108] A classic. Hausdorff and Packing measures on limit sets, Sullivan;s version of Patterson measure.

[106] First examples of groups whose limit sets have dimension 2.

[118] Tukia’s proof that geometrically finite limit sets have dimension < 2.

[120] Sharper gauge functions for geometrically finite limit sets.

[124] Extends Bowen’s results to some higher dimensional groups.

[82] Considers some infinitely generated groups.

Ergodic theory:

[90] Good reference for ergodic theory on limit sets. Explains much of Sullivan’s work in area.

Finiteness theorem:

[5] Ahlfors’ original paper.

[16] Bers version with estimates of area of quotient surface in terms of number of generators. Fixes error in original concerning triply punctured spheres.

[74] More on the finiteness theorem.

Geodesics:

[58] Good paper. Relates Poincaré exponent to bounded geodesics in quotient manifold. Covers all Fuchsian cases and geometrically finite Kleinian groups. Extended by [26].

[102] The Hausdorff dimension of bounded geodesics on geometrically finite Kleinian groups.

Harmonic measure:

[23] Characterizes when harmonic measure on two sides of a Jordan curve are mutually singular or mutually absolutely continuous.

[24] Generalizes F. and M. Riesz theorem to general simply connected domains. Solves Hayman-Wu problem.

[25] Simplifies and extends previous paper. Describes relation between Schwarzian derivative of Riemann mapping and the geometry of image domain.

[77] A classic. Proves Makarov’s law of the iterated logarithm for harmonic measure.

Got Salem prize for this and related results.

[84] the McMillan twist point theorem.

[98] Pommerenke’s improved version of Makarov’s upper bound for harmonic measure.

[123] Technical tour de force by Tom Wolff extending previous paper from simply connected domains to all planar domains.

Hausdorff dimension

[19] Gave alternate definition of Minkowski dimension without knowing it.

[60] Hausdorff dimension of continued fractions.

[116] [117] Alternate definitions of Minkowski dimension.

Hyperbolic 3-manifolds:

[30] A classic. Shows large class of groups are topologically tame.

[39] Explains relations between base eigenvalue, dimension and topological tameness. Highly recommended.

[40] Shows that topological tameness is equivalent to apparently stronger conditions geometric and analytic tameness. Proves that topologically tame limit sets are either whole sphere or zero area. Also highly recommended.

[87] Survey on Thurston’s uniformization theorem for three-dimensional manifolds.

[111] “The geometry and topology of 3-manifolds”. These are the unpublished (but widely referenced) notes of Thurston. They are available from the Geometry Center at the University of Minnesota. Send Email requesting them to admingeom.umn.edu you will get a copy and a (modest) bill.

Geometric finiteness:

[31] Gives many equivalent definitions of geometric finiteness.

[118] Geometrically finite limit sets have dimension < 2.

Heat equation: [46], [47], [49] Estimates for general manifolds and operators. [64], [63] Nice clean approach to decay estimates for the heat kernel.

Law of the iterated logarithm:

[11] [12] [9] [10] LIL’s for harmonic functions. [54], [72] Texts containing standard LIL for Brownian motion.

[20] Extensive survey of LIL’s in probability.

[42] LIL’s for martingales and harmonic functions. Results of this paper can be applied to Kleinian limits sets.

[77] LIL for harmonic measure.

[122] LIL for lacunary Fourier series.

Ahlfors measure conjecture:

[6] Ahlfors proof that limit sets of geometrically finite groups have zero measure.

[40] Canary‘s proof that topologically tame limit sets have zero area.

Mostow rigidity:

[88] Mostow’s paper. A classic.

[8] Astala-Zinsmeister counterexample for convergence groups.

[29] Sharper version of Mostow dichotomy for Fuchsian groups. Criterion for equivalence of groups.

[75] In Fuchsian case gives proof of Mostow dichotomy: conjugating map is either Möbius or singular.

[119] Stong version of Mostow’s theorem by Tukia.

Patterson measure:

[89] Results on limit set, conical limit set and δ. [93],[94] Patterson’s theory by Patterson.

[104] Sharp estimates for Patterson measure on geometrically finite groups with cusps.

[108] Sullivan’s version.

Pictures:

[86] Group theory applied to drawing limit sets.

[112] [113] [114] [115] Applications of Kleinian groups to physics? Nice pictures of limits sets. Also try the home pages on the world wide web of Curt McMullen at Berkeley (math.Berkeley.edu) and David Wright at Oklahoma State (math.okstate.edu).

Schottky groups:

[52] Good paper. Shows δ for classical Schottky groups is bounded away from 2. [35], [34] Appolonian packing groups, deformations.

Schwarzian derivative:

[53] Definition, basic properties, univalence. [24], [25] Relation to geometry of domain, arclength, tangents.

Survey papers:

[3] Survey by Abikoff of geometry of limit sets in 1988.

[18] Survey by Bers of Kleinian groups and Teichmüller theory in 1981.

[99] Explains Thurston’s approach to mortals.

[89] Results on limit set, conical limit set and δ.

[94] Patterson’s theory by Patterson.

[107] Sullivan’s paper “Discrete conformal groups and measurable dynamics”. Highly recommended.

[105] Sullivan’s paper “The density at infinity of a discrete groups of hyperbolic motions”. Another classic by a master.

Symmetric spaces:

[88] Definitions. [43], [44] Poincaré exponent equals dimension in some special cases for rank 1 symmetric space case.

[26] Poicaré equals dimension of conical limit set in all rank 1 cases.

Traveling Salesman:

[68] Gives explicit estimate for length of shortest curve connecting a given set of points.

[69] Early version of these results. Applications to Cauchy integral. Wiggly sets and support of harmonic measure.

[27] Uses results of [68] to estimate dimension of i limit sets.

[28] Applies ideas of previous paper to Brownian motion.

[91] Higher dimensional version of Jones result. Easier proof in 2 dimensions.

Uniformly perfect sets:

[38] Canary’s proof that limit sets are uniformly perfect.

[57] Great reference, discusses equivalent definitions.

[61] More definitions, particularly concerning covering group of complement.

[67] Uniformly perfect sets and quasiregular mappings.

[97] Pommerenke’s proof that limit sets are uniformly perfect.

Miscellaneous:

[21] Proof of Maskit’s conjecture on diameters of components from [80]. Alternate versions of Poincaré exponent.

[45] Constructing some odd 2-generator groups. [59] Shows Poincaré exponent strictly increases under combination of groups.

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