Primer on Mapping Class Groups (eBook)

Benson Farb, Dan Margalit (Autoren)

eBook Download: PDF | EPUB

2011
512 Seiten
Princeton University Press (Verlag)
978-1-4008-3904-9 (ISBN)

Infos

Benson Farb is professor of mathematics at the University of Chicago. He is the editor of Problems on Mapping Class Groups and Related Topics and the coauthor of Noncommutative Algebra. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology.

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The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmuller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Produkt-Details

• Erscheint lt. Verlag: 26.9.2011
• Reihe/Serie: Princeton Mathematical Series
• Reihe/Serie: Princeton Mathematical Series
• Zusatzinfo: 115 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: 3-manifold theory • Alexander method • algebraic integers • algebraic intersection number • algebraic relations • Algebraic Structure • annulus • aspherical manifold • automorphism • Basepoint • Benson Farb • bigon criterion • Big O notation • bijection • Birman exact sequence • BirmanЈilden theorem • braid group • branched cover • Calculation • capping homomorphism • Classification theorem • classifying space • Closed geodesic • Closed Surface • cohomology • collar lemma • Compactification (mathematics) • compactness criterion • Compact space • complex number • complex of curves • Computation • Configuration space • conjugacy class • continuous function • coordinates principle • coordinate system • corollary • Coset • Covering space • Curve • cutting homomorphism • Cyclic group • cyclic subgroup • Dehn twist • Dehn twists • DehnЌickorish theorem • DehnЎielsenЂaer theorem • Dennis Johnson • Diagram (category theory) • diffeomorphism • Dimension (vector space) • Disjoint sets • Disjoint union • Disk • Disk (mathematics) • Division by zero • Eigenvalues and Eigenvectors • Elementary matrix • equivalence class • equivalence relation • Euler characteristic • Euler class • exact sequence • existence theorem • existential quantification • extended mapping class group • FenchelЎielsen coordinates • fiber bundle • finite group • finite index • Finitely generated group • Finitely presented • finite-order homeomorphism • finite-order mapping class • finite subgroup • first homology group • foliation • free group • Fundamental domain • fundamental group • geodesic laminations • geometric classification • geometric group theory • geometric intersection number • geometric operation • Geometry • Gervais presentation • group action • Grtzsch's problem • harmonic maps • holomorphic quadratic differential • Homeomorphism • homological criterion • Homology (mathematics) • Homomorphism • Homotopy • Hyperbolic Geometry • hyperbolic plane • hyperbolic structure • Hyperbolic Surface • inclusion homomorphism • Infimum and supremum • infinity • intersection number • Intersection number (graph theory) • Intersection (set theory) • Isotopy • Johnson homomorphism • Jordan Curve Theorem • lantern relation • Linear map • Line (geometry) • low-dimensional homology • Mapping class group • mapping torus • Markov partitions • Mathematical Induction • measured foliations • measured foliation space • metric geometry • Metric Space • Meyer signature cocycle • Mod(S) • moduli space • Nielsen realization theorem • NielsenДhurston classification • NielsenДhurston classification theorem • normal subgroup • orbifold • Orbit • Orientability • outer automorphism group • Pair of pants (mathematics) • Parameter • Parity (mathematics) • Permutation • Pointwise • pseudo-Anosov homeomorphism • pullback • punctured disk • Putman • Quadratic differential • quasiconformal map • quasi-isometry • real number • rectangle • Riemannian manifold • Riemann surface • second homology group • simple closed curve • simplicial complex • Simply connected space • Special case • stretch factors • SUBGROUP • Subset • Summation • Surface • surface bundles • surface homeomorphism • Surjective function • symplectic representation • Tangent Space • Teichmller mapping • Teichmller metric • Teichmller space • Theorem • Thurston compactification • Topological space • Topology • Torelli group • Torsion • torus • train track • Train track (mathematics) • Transverse measure • uniformization theorem • Unit disk • Upper and lower bounds • Upper half-plane • Vector field • Vector Space • Wajnryb presentation • Word problem (mathematics)
• ISBN-10: 1-4008-3904-1 / 1400839041
• ISBN-13: 978-1-4008-3904-9 / 9781400839049