Flexibility of Group Actions on the Circle
In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety \mathrm{Hom}(L,\mathrm{Homeo}_+(S^1) contains uncountably many semi conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$ such that each representation lifts to \mathbb{R} For the case of most Fuchsian groups $L$ we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non elementary free or surface group admits an action on $S^1$ that is never semi conjugate to any action that factors through a finite dimensional connected Lie subgroup in \mathrm{Homeo}_+(S^1) It is exhibited that the mapping class groups of bounded surfaces have non semi conjugate faithful actions on $S^1$ In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of \mathrm{PSL}_2(\mathbb{R} which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of \mathbb{Z} valued subadditive defect one quasimorphisms on these groups would follow as a corollary. We also give a mostly self contained reconciliation of the various notions of semi conjugacy in the extant literature by showing that they are all equivalent.