Diffeomorphism group of the circle and discontinuous groups of theuniversal Teichmuller curve

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We studied a boundary of the smooth universal Teichmuller curve and the action of subgroups of diffeomorphism group of the circle on this boundary. Among actions of free product of two finite groups on the circle, we characterized Fuchsian actions in terms of rotation number.We obtained a result about simplicity of the reduced group C*-algebra of a group admitting convergence actions. We gave a necessary and sufficient condition for two geometrically finite convergence actions to admit an equivariant continuous map and applied it to construct examples of convergence actions.