A geometric Euler totient function associated to non-uniform lattices in SL(2,R)

We define a generalization of the Euler totient function associated to \Gamma, a subgroup of SL(2,\R) which is discrete, and whose quotient is non-compact but finite volume. When \Gamma = SL(2,Z) the generalization reduces to the classical Euler totient function. We will first discuss a counting result from the study of translation surfaces where the function arises. Next I will share an application of the counting result to understand a generalization of the Gauss circle problem, and propose further questions about the geometric Euler totient function.