The purpose of this volume and of the other volumes in the same series is to provide a collection of surveys that allows the reader to learn the important aspects of William Thurstons heritage. Thurstons ideas have altered the course of twentieth century mathematics, and they continue to have a significant influence on succeeding generations of mathematicians. The topics covered in the present volume include com-plex hyperbolic Kleinian groups, Möbius structures, hyperbolic ends, cone 3-manifolds, Thurstons norm, surgeries in representation varieties, triangulations, spaces of polygo-nal decompositions and of singular flat structures on surfaces, combination theorems in the theories of Kleinian groups, hyperbolic groups and holomorphic dynamics, the dynamics and iteration of rational maps, automatic groups, and the combinatorics of right-angled Artin groups.

Kenichi Ohshika is a professor of mathematics at Gakushuin University, and a professor emeritus at Osaka University. He receivedhis PhD from the University of Tokyo in 1989, and worked at Tokyo Institute of Technology, University of Tokyo, and Osaka University before joining Gakushuin University. His interests includes Kleinian groups, hyperbolic geometry, Teichmüller theory, and 3-manifolds. He has held visiting positions at Institut des Hautes Études Scientifiques (1990-1991, 1999-2000), University of Warwick (1988-89, 1993, 2007), Université de Montréal (1995), Hausdorff Research Institute of Mathematics (2010), Université de Strasbourg (2014, 2019) and the Korea Institute of Advanced Science (2009-2015). He is a recipient of the MSJ geometry prize (2012), and the author of more than 60 papers.

Athanase Papadopoulos (born 1957) is Directeur de Recherche at the French Centre National de la Recherche Scientifique. His mainfields of interest are geometry and topology, the history and philosophy of mathematics, and mathematics and music. He has held visiting positions at the Institute for Advanced Study, Princeton (198485 and 199394), USC (19981999), CUNY (Ada Peluso Professor, 2014), Brown University (Distinguished visiting professor, 2017), Tsinghua University, Beijing (2018), Lamé Chair of the State University of Saint Petersburg (2019), and has had several month visits to the Max-Plank Institute for mathematics (Bonn), the Erwin Schrödinger Institute (Vienna), the Graduate Center of CUNY (New York), the Tata Institute (Bombay), Galatasaray University (Istanbul), the University of Florence (Italy), Fudan University (Shanghai), Gakushuin University (Tokyo) and Presidency University (Calcutta). He is the author of more than 200 published articles and 35 monographs and edited books. 

1 Ken'ichi Ohshika and Athanase Papadopoulos, Introduction.- 2 Michael Kapovich, A survey of complex hyperbolic Kleinian groups.- 3 Graham Smith, Möbius structures, hyperbolic ends and K-surfaces in hyperbolic space.- 4 Joan Porti, Cone 3-manifods.- 5 Takahiro Kitayama, A survey of the Thurston norm.- 6 Georgios Kydonakis, From hyperbolic Dehn filling to surgeries in representation varieties.- 7 Sang-Hyun Kim, Acute geodesic triangulations of manifolds.- 8 Ismail Salam, Signature calculation of the area Hermitian form on some spaces of polygons.- 9 Mikhail Chernaviskikh, Altan Erdnigor, Nikita Kalinin, and Alexandr Zakharov, Equilateral convex triangulations of RP2 with three conical points of equal defect.- 10 Mahan Mj and Sabyasachi Mukherjee, Combination theorems in groups, geometry and dynamics.- 11 Kevin Pilgrim, On the pullback relation on curves induced by a Thurston map.- 12 William Floyd, The pullback map on Teichmüller space induced from a Thurston map.- 13 Russell Lodge, Yauhen Mikulich and Dierk Schleicher, A classification of postcritically finite Newton maps.- 14 Sarah Rees, The development of the theory of automatic groups.- 15 Thomas Koberda, Geometry and combinatorics via right-angled Artin groups.