Index of Γ-Equivariant Toeplitz Operators

Let Γ be a discrete subgroup of PSL(2,ℝ) of infinite covolume with infinite conjugacy classes. H_t denotes the Hilbert space consisting of analytic functions in $${L^2}(\mathbb{D},{({\text{Im }}z)^{t-2}}{\text{d}}\bar z{\text{d}}z)$$ and, for t > 1, π_t denotes the corresponding projective unitary representation of PSL(2, ℝ) on this Hilbert space. Let A_t be the II∞ factor given by the commutant of π_t(Γ) in B(H_t). Let F denote a fundamental domain for Γ in D. We assume that t > 5 and give $$partial M = \partial \mathbb{D} \cap \bar F$$ the topology of disjoint union of its connected components.