Let M be a circle or a compact interval, and let α " k`τ ě 1 be a real number such that k " tαu. We write Diff ὰ pMq for the group of orientation preserving C k diffeomorphisms of M whose k th derivatives are Hölder continuous with exponent τ. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G ď Diff ὰ pMq with the property that G admits no injective homomorphisms into Ť βąα Diff β pMq. We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of Ş βăα Diff β pMq with the property that G admits no injective homomorphisms into Diff ὰ pMq. The groups G are constructed so that their commutator groups are simple. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if α ě 1 is a real number not equal to 2, then there is no nontrivial homomorphism Diff ὰ pS 1 q Ñ Ť βąα Diff β pS 1 q. Finally, we obtain an independent result that the class of finitely generated subgroups of Diff 1 pMq is not closed under taking finite free products.

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 HompL, Homeo`pS 1 qq 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 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 Homeo`pS 1 q. 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 PSL 2 pRq 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 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.

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