Flexibility of Group Actions on the Circle

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.

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.