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The goal of this Master thesis is to study several properties of dendric sets or words, which generalise Arnoux-Rauzy words and codings of regular interval exchanges. These properties mainly focus on prefix, suffix and bifix codes included in such a set and on the free group. One of the main results states that, in a recurrent dendric set, the set of (first) return words of any word is a basis of the free group. Using this result and various graphs and automata, I study several properties which then give me, among other results, the stability of the family of recurrent dendric sets under bifix decoding. Finally, I briefly introduce S-adic representations and give a particular representation for recurrent dendric words. Ce mémoire a pour but d'étudier différentes propriétés des ensembles ou mots dendriques qui sont une généralisation des mots d'Arnoux-Rauzy et des codages d'intervalles réguliers. Ces propriétés sont essentiellement centrées autour des codes préfixes, suffixes et bifixes inclus dans un tel ensemble et autour de la notion de groupe libre. Un des résultats majeurs affirme que dans un ensemble dendrique récurrent, les mots de (premier) retour pour n'importe quel mot forment une base du groupe libre. En utilisant des graphes et des automates particuliers, nous étudions ensuite diverses propriétés qui nous permettent entre autres d'obtenir la stabilité de la famille des ensembles dendriques récurrents pour l'opération de décodage bifixe. Enfin, nous introduisons brièvement la notion de représentation S-adique et donnons une représentation particulière pour les mots dendriques récurrents.

discrete mathematics --- words --- dendric --- bifix code --- free group --- mathématiques discrètes --- mots --- dendrique --- code bifixe --- groupe libre --- Physique, chimie, mathématiques & sciences de la terre > Mathématiques

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Mathematical analysis --- Free products (Group theory) --- Geometric group theory --- HNN-extensions --- Extensions, Higman-Neumann-Neumann --- Extensions, HNN --- -Higman-Neumann-Neumann extensions --- Group extensions (Mathematics) --- Group theory --- Products, Free (Group theory) --- Extensions HNN --- Groupes, Théorie géométrique des --- Groupes, Théorie des --- Extensions HNN. --- Groupes, Théorie géométrique des. --- Groupes, Théorie des.

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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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In recent years, considerable progress has been made in studying algebraic cycles using infinitesimal methods. These methods have usually been applied to Hodge-theoretic constructions such as the cycle class and the Abel-Jacobi map. Substantial advances have also occurred in the infinitesimal theory for subvarieties of a given smooth variety, centered around the normal bundle and the obstructions coming from the normal bundle's first cohomology group. Here, Mark Green and Phillip Griffiths set forth the initial stages of an infinitesimal theory for algebraic cycles. The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.

512.73 --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic cycles. --- Hodge theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Cycles, Algebraic --- Algebraic cycles --- Hodge theory --- Addition. --- Algebraic K-theory. --- Algebraic character. --- Algebraic curve. --- Algebraic cycle. --- Algebraic function. --- Algebraic geometry. --- Algebraic number. --- Algebraic surface. --- Algebraic variety. --- Analytic function. --- Approximation. --- Arithmetic. --- Chow group. --- Codimension. --- Coefficient. --- Coherent sheaf cohomology. --- Coherent sheaf. --- Cohomology. --- Cokernel. --- Combination. --- Compass-and-straightedge construction. --- Complex geometry. --- Complex number. --- Computable function. --- Conjecture. --- Coordinate system. --- Coprime integers. --- Corollary. --- Cotangent bundle. --- Diagram (category theory). --- Differential equation. --- Differential form. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Divisor. --- Duality (mathematics). --- Elliptic function. --- Embedding. --- Equation. --- Equivalence class. --- Equivalence relation. --- Exact sequence. --- Existence theorem. --- Existential quantification. --- Fermat's theorem. --- Formal proof. --- Fourier. --- Free group. --- Functional equation. --- Generic point. --- Geometry. --- Group homomorphism. --- Hereditary property. --- Hilbert scheme. --- Homomorphism. --- Injective function. --- Integer. --- Integral curve. --- K-group. --- K-theory. --- Linear combination. --- Mathematics. --- Moduli (physics). --- Moduli space. --- Multivector. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Open problem. --- Parameter. --- Polynomial ring. --- Principal part. --- Projective variety. --- Quantity. --- Rational function. --- Rational mapping. --- Reciprocity law. --- Regular map (graph theory). --- Residue theorem. --- Root of unity. --- Scientific notation. --- Sheaf (mathematics). --- Smoothness. --- Statistical significance. --- Subgroup. --- Summation. --- Tangent space. --- Tangent vector. --- Tangent. --- Terminology. --- Tetrahedron. --- Theorem. --- Transcendental function. --- Transcendental number. --- Uniqueness theorem. --- Vector field. --- Vector space. --- Zariski topology.

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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