Handbook of Teichmüller theory, Volume III

Athanase Papadopoulos (1)

1 IRMA - Institut de Recherche Mathématique Avancée

The subject of this handbook is Teichmüller theory in a wide sense, namely the theory of geometric structures on surfaces and their moduli spaces. This includes the study of vector bundles on these moduli spaces, the study of mapping class groups, the relation with 3-manifolds, the relation with symmetric spaces and arithmetic groups, the representation theory of fundamental groups, and applications to physics. Thus the handbook is a place where several fields of mathematics interact: Riemann surfaces, hyperbolic geometry, partial differential equations, several complex variables, algebraic geometry, algebraic topology, combinatorial topology, low-dimensional topology, theoretical physics, and others. This confluence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics. The present volume contains surveys on the fundamental theory as well as surveys on applications to and relations with the fields mentioned above. It is written by leading experts in the fields. Some of the surveys contain classical material, while others present the latest developments of the theory as well as open problems. This volume includes the following four sections: (1) The metric and the analytic theory. (2) The group theory. (3) The algebraic topology of mapping class groups and moduli spaces. (4) Teichmüller theory and mathematical physics. The handbook is addressed to graduate students and researchers in all the fields mentioned.

Mots clés: exchange automorphism; earthquake curves; infinitesimal earthquakes; cross-ratio distortions and Zygmund bounded functions; Arithmetic groups; symmetric spaces; Teichmüller spaces; Lie groups; transformation groups; proper actions; classifying spaces; locally symmetric spaces; Riemann surfaces; reduction theories; fundamental domains; quantum theory; compactifications; boundaries; universal spaces; duality groups; curve complexes; Tits buildings; hyperbolic groups; Coxeter groups; outer automorphism groups; Schottky problems; pants decompositions; Generalized Mummford conjecture; Mumford-Morita-Miller classes; Pontryagin-Thom cobordism theory; Mumford conjecture; classifying space; rational cohomology of moduli space; Line of minima; Teichmüller space; geodesic lamination; earthquake; Teichmüller geodesic; quasifuchsian group; Teichmülller theory; moduli spaces; mapping class groups; 3-manifolds; Beltrami equation; uniformization; quasiconformal map; Finite earthquakes; earthquakes; earthquake measures; quasisymmetric and symmetric homeomorphisms; Simplicial complex; simplicial automorphism; Mapping class group; Thompson group; Ptolemy groupoid; infinite braid group; quantization; braided Thompson group; Euler class; discrete Godbillon-Vey class; Hatcher-Thurston complex; combable group; finitely presented group; central extension; Grothendieck-Teichmüller group; renormalized volume; hyperbolic manifolds; Weil-Petersson metric; Liouville equation; surface; curve complex; arc complex; arc and curve complex; boundary graph complex; complex of nonseparating curves; complex of separating curves; ideal triangulation; complex of domains; truncate complex of domains; pants decomposition; cut system; Torelli complex

Domaines: Topologie géométrique [math.GT]

Athanase Papadopoulos (Dir.). Handbook of Teichmüller theory, Volume III. European Mathematical Society, pp.874, 2012, 978-3-03719-103-3. ⟨10.4171/103⟩. ⟨hal-00706331⟩