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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors.An essential primer for undergraduates making the leap to graduate work, the book begins with free groups-actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples.Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Geometric group theory. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem.

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This book has as its subject the boundary value theory of holomorphic functions in several complex variables, a topic that is just now coming to the forefront of mathematical analysis. For one variable, the topic is classical and rather well understood. In several variables, the necessary understanding of holomorphic functions via partial differential equations has a recent origin, and Professor Stein's book, which emphasizes the potential-theoretic aspects of the boundary value problem, should become the standard work in the field.Originally published in 1972.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Mathematical potential theory --- Holomorphic functions --- Harmonic functions --- Holomorphic functions. --- Harmonic functions. --- Fonctions de plusieurs variables complexes. --- Functions of several complex variables --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Functions, Holomorphic --- Absolute continuity. --- Absolute value. --- Addition. --- Ambient space. --- Analytic function. --- Arbitrarily large. --- Bergman metric. --- Borel measure. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Boundedness. --- Brownian motion. --- Calculation. --- Change of variables. --- Characteristic function (probability theory). --- Combination. --- Compact space. --- Complex analysis. --- Complex conjugate. --- Computation. --- Conformal map. --- Constant term. --- Continuous function. --- Coordinate system. --- Corollary. --- Cramer's rule. --- Determinant. --- Diameter. --- Dimension. --- Elliptic operator. --- Estimation. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Fatou's theorem. --- Function space. --- Green's function. --- Green's theorem. --- Haar measure. --- Half-space (geometry). --- Harmonic function. --- Hilbert space. --- Holomorphic function. --- Hyperbolic space. --- Hypersurface. --- Hölder's inequality. --- Invariant measure. --- Invertible matrix. --- Jacobian matrix and determinant. --- Line segment. --- Linear map. --- Lipschitz continuity. --- Local coordinates. --- Logarithm. --- Majorization. --- Matrix (mathematics). --- Maximal function. --- Measure (mathematics). --- Minimum distance. --- Natural number. --- Normal (geometry). --- Open set. --- Order of magnitude. --- Orthogonal complement. --- Orthonormal basis. --- Parameter. --- Poisson kernel. --- Positive-definite matrix. --- Potential theory. --- Projection (linear algebra). --- Quadratic form. --- Quantity. --- Real structure. --- Requirement. --- Scientific notation. --- Sesquilinear form. --- Several complex variables. --- Sign (mathematics). --- Smoothness. --- Subgroup. --- Subharmonic function. --- Subsequence. --- Subset. --- Summation. --- Tangent space. --- Theorem. --- Theory. --- Total variation. --- Transitive relation. --- Transitivity. --- Transpose. --- Two-form. --- Unit sphere. --- Unitary matrix. --- Vector field. --- Vector space. --- Volume element. --- Weak topology.

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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors.An essential primer for undergraduates making the leap to graduate work, the book begins with free groups-actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples.Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Geometric group theory. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem.

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This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

Topology --- Differential geometry. Global analysis --- Geometry, Hyperbolic --- Three-manifolds (Topology) --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- Geometry, Hyperbolic. --- 514.1 --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- General geometry --- Three-manifolds (Topology). --- 514.1 General geometry --- Géométrie hyperbolique --- Variétés topologiques à 3 dimensions --- 3-sphere. --- Abelian group. --- Affine space. --- Affine transformation. --- Atlas (topology). --- Automorphism. --- Basis (linear algebra). --- Bounded set (topological vector space). --- Brouwer fixed-point theorem. --- Cartesian coordinate system. --- Characterization (mathematics). --- Compactification (mathematics). --- Conformal map. --- Contact geometry. --- Curvature. --- Cut locus (Riemannian manifold). --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Disk (mathematics). --- Divisor (algebraic geometry). --- Dodecahedron. --- Eigenvalues and eigenvectors. --- Embedding. --- Euclidean space. --- Euler number. --- Exterior (topology). --- Facet (geometry). --- Fiber bundle. --- Foliation. --- Fundamental group. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Half-space (geometry). --- Holonomy. --- Homeomorphism. --- Homotopy. --- Horocycle. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hyperboloid model. --- Interior (topology). --- Intersection (set theory). --- Isometry group. --- Isometry. --- Jordan curve theorem. --- Lefschetz fixed-point theorem. --- Lie algebra. --- Lie group. --- Line (geometry). --- Linear map. --- Linearization. --- Manifold. --- Mathematical induction. --- Metric space. --- Moduli space. --- Möbius transformation. --- Norm (mathematics). --- Pair of pants (mathematics). --- Piecewise linear manifold. --- Piecewise linear. --- Poincaré disk model. --- Polyhedron. --- Projection (linear algebra). --- Projection (mathematics). --- Pseudogroup. --- Pullback (category theory). --- Quasi-isometry. --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann surface. --- Riemannian manifold. --- Sheaf (mathematics). --- Sign (mathematics). --- Simplicial complex. --- Simply connected space. --- Special linear group. --- Stokes' theorem. --- Subgroup. --- Subset. --- Tangent space. --- Tangent vector. --- Tetrahedron. --- Theorem. --- Three-dimensional space (mathematics). --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Transversal (geometry). --- Two-dimensional space. --- Uniformization theorem. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Topologie algébrique --- Topologie combinatoire --- Algebraic topology. --- Combinatorial topology. --- Variétés topologiques --- Geometrie --- Theorie des noeuds

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Many parallels between complex dynamics and hyperbolic geometry have emerged in the past decade. Building on work of Sullivan and Thurston, this book gives a unified treatment of the construction of fixed-points for renormalization and the construction of hyperbolic 3- manifolds fibering over the circle. Both subjects are studied via geometric limits and rigidity. This approach shows open hyperbolic manifolds are inflexible, and yields quantitative counterparts to Mostow rigidity. In complex dynamics, it motivates the construction of towers of quadratic-like maps, and leads to a quantitative proof of convergence of renormalization.

Differential dynamical systems --- Drie-menigvuldigheden (Topologie) --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Differentiable dynamical systems. --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Low-dimensional topology --- Topological manifolds --- Algebraic topology. --- Analytic continuation. --- Automorphism. --- Beltrami equation. --- Bifurcation theory. --- Boundary (topology). --- Cantor set. --- Circular symmetry. --- Combinatorics. --- Compact space. --- Complex conjugate. --- Complex manifold. --- Complex number. --- Complex plane. --- Conformal geometry. --- Conformal map. --- Conjugacy class. --- Convex hull. --- Covering space. --- Deformation theory. --- Degeneracy (mathematics). --- Dimension (vector space). --- Disk (mathematics). --- Dynamical system. --- Eigenvalues and eigenvectors. --- Factorization. --- Fiber bundle. --- Fuchsian group. --- Fundamental domain. --- Fundamental group. --- Fundamental solution. --- G-module. --- Geodesic. --- Geometry. --- Harmonic analysis. --- Hausdorff dimension. --- Homeomorphism. --- Homotopy. --- Hyperbolic 3-manifold. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypersurface. --- Infimum and supremum. --- Injective function. --- Intersection (set theory). --- Invariant subspace. --- Isometry. --- Julia set. --- Kleinian group. --- Laplace's equation. --- Lebesgue measure. --- Lie algebra. --- Limit point. --- Limit set. --- Linear map. --- Mandelbrot set. --- Manifold. --- Mapping class group. --- Measure (mathematics). --- Moduli (physics). --- Moduli space. --- Modulus of continuity. --- Möbius transformation. --- N-sphere. --- Newton's method. --- Permutation. --- Point at infinity. --- Polynomial. --- Quadratic function. --- Quasi-isometry. --- Quasiconformal mapping. --- Quasisymmetric function. --- Quotient space (topology). --- Radon–Nikodym theorem. --- Renormalization. --- Representation of a Lie group. --- Representation theory. --- Riemann sphere. --- Riemann surface. --- Riemannian manifold. --- Schwarz lemma. --- Simply connected space. --- Special case. --- Submanifold. --- Subsequence. --- Support (mathematics). --- Tangent space. --- Teichmüller space. --- Theorem. --- Topology of uniform convergence. --- Topology. --- Trace (linear algebra). --- Transversal (geometry). --- Transversality (mathematics). --- Triangle inequality. --- Unit disk. --- Unit sphere. --- Upper and lower bounds. --- Vector field. --- Differentiable dynamical systems --- 515.16 --- 515.16 Topology of manifolds --- Topology of manifolds