  We compute the C*-algebra generated by a group of composition operators
acting on certain reproducing kernel Hilbert spaces over the disk, where the
symbols belong to a non-elementary Fuchsian group. We show that such a
C*-algebra contains the compact operperators, and its quotient is isomorphic to
the crossed product C*-algebra determined by the action of the group on the
boundary circle. In addition we show that the C*-algebras obtained from
composition operators acting on a natural family of Hilbert spaces are in fact
isomorphic, and also determine the same Ext-class, which can be related to
known extensions of the crossed product.
