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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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The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology.In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems.Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.

Braid theory --- Braids, Theory of --- Theory of braids --- Braid theory. --- Algebraic topology --- Knot theory --- Representations of groups --- 512.54 --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Knots (Topology) --- Low-dimensional topology --- 512.54 Groups. Group theory --- Groups. Group theory --- Knot theory. --- Representations of groups. --- Addition. --- Alexander polynomial. --- Algebraic structure. --- Automorphism. --- Ball (mathematics). --- Bijection. --- Braid group. --- Branched covering. --- Burau representation. --- Calculation. --- Cartesian coordinate system. --- Characterization (mathematics). --- Coefficient. --- Combinatorial group theory. --- Commutative property. --- Commutator subgroup. --- Configuration space. --- Conjugacy class. --- Corollary. --- Covering space. --- Dehn twist. --- Determinant. --- Diagram (category theory). --- Dimension. --- Disjoint union. --- Double coset. --- Eigenvalues and eigenvectors. --- Enumeration. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Faithful representation. --- Finite set. --- Free abelian group. --- Free group. --- Fundamental group. --- Geometry. --- Group (mathematics). --- Group ring. --- Groupoid. --- Handlebody. --- Heegaard splitting. --- Homeomorphism. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity element. --- Identity matrix. --- Inclusion map. --- Initial point. --- Integer matrix. --- Integer. --- Knot polynomial. --- Lens space. --- Line segment. --- Line–line intersection. --- Link group. --- Low-dimensional topology. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix group. --- Matrix representation. --- Monograph. --- Morphism. --- Natural transformation. --- Normal matrix. --- Notation. --- Orientability. --- Parity (mathematics). --- Permutation. --- Piecewise linear. --- Pointwise. --- Polynomial. --- Prime knot. --- Projection (mathematics). --- Proportionality (mathematics). --- Quotient group. --- Requirement. --- Rewriting. --- Riemann surface. --- Semigroup. --- Sequence. --- Special case. --- Subgroup. --- Submanifold. --- Subset. --- Symmetric group. --- Theorem. --- Theory. --- Topology. --- Trefoil knot. --- Two-dimensional space. --- Unimodular matrix. --- Unit vector. --- Variable (mathematics). --- Word problem (mathematics). --- Topologie algébrique

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William Thurston (1946–2012) was one of the great mathematicians of the twentieth century. He was a visionary whose extraordinary ideas revolutionized a broad range of mathematical fields, from foliations, contact structures, and Teichmüller theory to automorphisms of surfaces, hyperbolic geometry, geometrization of 3-manifolds, geometric group theory, and rational maps. In addition, he discovered connections between disciplines that led to astonishing breakthroughs in mathematical understanding as well as the creation of entirely new fields. His far-reaching questions and conjectures led to enormous progress by other researchers. What's Next? brings together many of today's leading mathematicians to describe recent advances and future directions inspired by Thurston's transformative ideas.Including valuable insights from his colleagues and former students, What's Next? discusses Thurston's fundamental contributions to topology, geometry, and dynamical systems and includes many deep and original contributions to the field. This incisive and wide-ranging book also explores how he introduced new ways of thinking about and doing mathematics, innovations that have had a profound and lasting impact on the mathematical community as a whole.

Dynamics. --- Geometry. --- Topology. --- MATHEMATICS / General. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Mathematics --- Euclid's Elements --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Thurston, William P., --- Thurston, W. P. --- Arbitrarily large. --- Asymptotic expansion. --- Automorphism. --- Big O notation. --- Braid group. --- Branch point. --- Central series. --- Character variety. --- Characterization (mathematics). --- Cohomology operation. --- Cohomology. --- Commutative property. --- Conjecture. --- Conjugacy class. --- Convex hull. --- Covering space. --- Coxeter group. --- Curvature. --- Dehn's lemma. --- Diagram (category theory). --- Disjoint union. --- Eigenfunction. --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Extreme point. --- Faithful representation. --- Fiber bundle. --- Free group. --- Free product. --- Fundamental group. --- Geometrization conjecture. --- HNN extension. --- Haar measure. --- Homeomorphism. --- Homotopy. --- Hyperbolic 3-manifold. --- Hyperbolic geometry. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypercube. --- I0. --- Inclusion map. --- Incompressible surface. --- JSJ decomposition. --- Jordan curve theorem. --- Julia set. --- Klein bottle. --- Kleinian group. --- Lebesgue measure. --- Leech lattice. --- Limit point. --- Lyapunov exponent. --- Mahler measure. --- Manifold decomposition. --- Mapping cylinder. --- Marriage theorem. --- Maxima and minima. --- Moduli space. --- Möbius strip. --- Möbius transformation. --- Natural topology. --- Non-Euclidean geometry. --- Non-positive curvature. --- Normal subgroup. --- Open set. --- Orientability. --- Pair of pants (mathematics). --- Perfect group. --- Pleated surface. --- Polynomial. --- Preorder. --- Probability measure. --- Pullback (category theory). --- Pullback (differential geometry). --- Quadric. --- Quasi-isometry. --- Quasiconvex function. --- Rectangle. --- Riemann surface. --- Riemannian manifold. --- Saddle point. --- Sectional curvature. --- Sign (mathematics). --- Simple algebra. --- Simply connected space. --- Special case. --- Subgroup. --- Subset. --- Symplectic geometry. --- Theorem. --- Total order. --- Unit disk. --- Unit sphere. --- Upper and lower bounds. --- Vector bundle.

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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.