$C*$-algebras generated by groups of composition operators Michael Jury 47B33composition operatorC*-algebra 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 operators, 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. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3164 10.1512/iumj.2007.56.3164 en Indiana Univ. Math. J. 56 (2007) 3171 - 3192 state-of-the-art mathematics http://iumj.org/access/