Flexibility of Group Actions on the Circle: (2231 Lecture Notes in Mathematics)

In this partly expository work, a framework is developed for building exotic circle actions of certain classical groups. The authors give general combination theorems for indiscrete isometry groups of hyperbolic space which apply to Fuchsian and limit groups. An abundance of integer-valued subadditive defect-one quasimorphisms on these groups follow as a corollary. The main classes of groups considered are limit and Fuchsian groups. Limit groups are shown to admit large collections of faithful actions on the circle with disjoint rotation spectra. For Fuchsian groups, further flexibility results are proved and the existence of non-geometric actions of free and surface groups is established. An account is given of the extant notions of semi-conjugacy, showing they are equivalent. This book is suitable for experts interested in flexibility of representations, and for non-experts wanting an introduction to group representations into circle homeomorphism groups.

Table of Contents:
- Introduction.- Preliminaries.- Topological Baumslag Lemmas. - Splittable Fuchsian Groups. - Combination Theorem for Flexible Groups. - Axiomatics. - Mapping Class Groups. - Zero Rotation Spectrum and Teichmüller Theory.

About the Author :
Sang-hyun Kim received his PhD from Yale University in 2007, under the supervision of Andrew J. Casson. He previously worked at the University of Texas at Austin, Tufts University and KAIST. He is a member of the Young Korean Academy of Science and Technology (Y-KAST), and a recipient of the Sang-San Prize for Young Mathematicians. He is currently an Associate Professor at Seoul National University. Thomas Koberda received his PhD in 2012 from Harvard University, under the supervision of Curtis T. McMullen. He was an NSF Postdoctoral Fellow and a Gibbs Assistant Professor at Yale University. He is currently on the faculty of the University of Virginia. In 2017, he was named an Alfred P. Sloan Foundation Research Fellow, and was awarded the Kamil Duszenko Prize. Mahan Mj received his PhD from the University of California at Berkeley in 1997, under the supervision of Andrew J. Casson. He is currently Professor of Mathematics at Tata Institute of Fundamental Research. He was awarded the Infosys Prize in Mathematical Sciences in 2015 and was an invited speaker in the Geometry Section at the International Congress of Mathematicians, 2018.

Review :
“The book contains a lot of information and is written in a concise and well-organized way, describing its contents in a long introductory chapter and starting each chapter with a small abstract, with many references to the relevant literature and comments on related concepts and on related work (maybe sometimes the addition of more intuitive versions and motivations of some of the basic, sometimes quite technical definitions and concepts would have been helpful for a less experienced reader).” (Bruno Zimmermann, zbMATH 1407.57001, 2019)