Counting Resonances on Hyperbolic Surfaces with Unitary Twists

We present the Laplace operator associated to a hyperbolic surface $\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental group $\Gamma$, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of $\mathbb{C}$ by constructing a parametrix for the Laplacian, following the approach by Guillop\'e and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.