  Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and
let M denote the quotient of the unit disc by $\Gamma$. We prove that a
Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm
if its symbol is invertible on the boundary of M and its Brauer index is equal
to the winding number of f at the boundary. We construct the associated
extension of the algebra of functions continuous on the boundary of M by the
Brauer ideal in the C*-algebra generated by such operators.
