ar X iv : m at h / 99 11 04 2 v 1 [ m at h . O A ] 8 N ov 1 99 9 INDEX OF Γ - EQUIVARIANT TOEPLITZ OPERATORS

Let Γ be a discrete subgroup of PSL(2,R) of infinite covolume with infinite conjugacy classes. Let Ht be the Hilbert space consisting of analytic functions in L(D, (Im z)dzdz) and let, for t > 1, πt denote the corresponding projective unitary representation of PSL(2,R) on this Hilbert space. We denote by At the II∞ factor given by the commutant of πt(Γ) in B(Ht). Let F denote a fundamental domain for Γ in D and assume that t > 5. ∂M = ∂D ∩ F is given the topology of disjoint union of its connected components. Suppose that f is a continuous Γ-invariant function on D whose restriction to F extends to a continuous function on F and such that f |∂M is an invertible element of C0(∂M) ̃. Let T t f = PtMfPt denote the Toeplitz operator with symbol f . Then T t f is Fredholm, in the Breuer sense, with respect to the II∞ factor At and, moreover, its Breuer index is equal to the total winding number of f on ∂M .