Rotation spectra and exotic group actions on the circle

We develop a general framework for producing uncountable families of exotic actions of certain classically studied groups on the circle. We show that if G is a closed surface group then there are connected components of the nonlinear representation variety HompG,Homeo`pS 1qq which contain uncountably many semi-conjugacy classes of G–actions, thus complementing a result of K. Mann. We also exhibit non-semi-conjugate actions of mapping class groups of surfaces with boundary on the circle. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of PSL2pRq as well as a topological Baumslag Lemma. As a corollary, we show that for many of the groups, there exist uncountably many integer-valued subadditive quasimorphisms which are defect–one and which are linearly independent in H2 bpG;Zq. We also give a mostly self-contained reconciliation of the various notions of semiconjugacy in the extant literature by showing that they are all equivalent.