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This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids "ations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

CR submanifolds. --- Dehn surgery (Topology). --- Three-manifolds (Topology). --- CR submanifolds --- Dehn surgery (Topology) --- Three-manifolds (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Low-dimensional topology --- Topological manifolds --- Surgery (Topology) --- Manifolds (Mathematics) --- Arc (geometry). --- Automorphism. --- Ball (mathematics). --- Bijection. --- Bump function. --- CR manifold. --- Calculation. --- Canonical basis. --- Cartesian product. --- Clifford torus. --- Combinatorics. --- Compact space. --- Conjugacy class. --- Connected space. --- Contact geometry. --- Convex cone. --- Convex hull. --- Coprime integers. --- Coset. --- Covering space. --- Dehn surgery. --- Dense set. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differential geometry of surfaces. --- Discrete group. --- Double coset. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Equivalence relation. --- Euclidean distance. --- Four-dimensional space. --- Function (mathematics). --- Fundamental domain. --- Geometry and topology. --- Geometry. --- Harmonic function. --- Hexagonal tiling. --- Holonomy. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Horosphere. --- Hyperbolic 3-manifold. --- Hyperbolic Dehn surgery. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperbolic triangle. --- Hypersurface. --- I0. --- Ideal triangle. --- Intermediate value theorem. --- Intersection (set theory). --- Isometry group. --- Isometry. --- Limit point. --- Limit set. --- Manifold. --- Mathematical induction. --- Metric space. --- Möbius transformation. --- Parameter. --- Parity (mathematics). --- Partial derivative. --- Partition of unity. --- Permutation. --- Polyhedron. --- Projection (linear algebra). --- Projectivization. --- Quotient space (topology). --- R-factor (crystallography). --- Real projective space. --- Right angle. --- Sard's theorem. --- Seifert fiber space. --- Set (mathematics). --- Siegel domain. --- Simply connected space. --- Solid torus. --- Special case. --- Sphere. --- Stereographic projection. --- Subgroup. --- Subsequence. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Topology. --- Torus. --- Transversality (mathematics). --- Triangle group. --- Union (set theory). --- Unit disk. --- Unit sphere. --- Unit tangent bundle.

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Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. The Plaid Model, which is a self-contained sequel to Richard Schwartz's Outer Billiards on Kites, provides a combinatorial model for orbits of outer billiards on kites.Schwartz relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called "the plaid model," has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics.The book includes an extensive computer program that allows readers to explore materials interactively and each theorem is accompanied by a computer demo.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- 1-dimensional wordlines. --- Bad Tile Lemma. --- Box Theorem. --- Copy Lemma. --- Copy Theorem. --- Curve Turning Theorem. --- Graph Master Picture Theorem. --- Graph Master Theorem. --- Graph Reconstruction Lemma. --- Grid Geometry Lemma. --- Grid Supply Lemma. --- Horizontal Lemma. --- Intertwining Lemma. --- Master Picture Theorem. --- Matching Criterion. --- Orbit Equivalence Theorem. --- PET. --- Plaid Master Picture Theorem. --- Projection Theorem. --- Quasi-Isomorphism Theorem. --- Rectangle Lemma. --- Renormalization Theorem. --- Segment Lemma. --- Truchet Comparison Theorem. --- Truchet tile system. --- Vertical Lemma. --- anchor point. --- arithmetic alignment. --- arithmetic graph. --- auxiliary lemmas. --- capacities. --- capacity sequence. --- capacity. --- checkerboard partition. --- classifying map. --- compactification. --- congruence. --- continued fractions. --- convex polygon. --- convex polytopes. --- corner percolation. --- east edges. --- elementary number theory. --- embedded loops. --- equidistribution. --- even rational parameter. --- fundamental surface. --- geometric alignment. --- geometry. --- graph grid. --- grid lines. --- horizontal case. --- integer square. --- intersection points. --- kites. --- kits. --- lemma. --- lemmas. --- light point. --- light points. --- linear algebra. --- low capacity lines. --- map. --- maps. --- mass sequence. --- masses. --- number theory. --- orbit. --- oriented lines. --- outer billiards map. --- outer billiards. --- parallel lines. --- parallelotope. --- particle lines. --- pixelated spacetime diagrams. --- pixelated spacetime. --- pixilation. --- plaid PET map. --- plaid model. --- plaid polygon. --- plaid polygons. --- plane. --- planetary motion. --- planetary orbits. --- polygon. --- polytope exchange transformations. --- polytopes. --- prism structure. --- prism. --- proof. --- quarter turn compositions. --- remote adjacency. --- renormalization. --- scale information. --- sequences. --- slanting lines. --- spaces. --- spacetime diagrams. --- spacetime plaid surfaces. --- special billiards orbits. --- square tiling. --- stacking blocks. --- structural result. --- symmetries. --- symmetry. --- technical lemma. --- theorems. --- vertical case. --- vertical particles.

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Factor tables --- Numbers, Prime --- Counting

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