Counting resonances, in the sense of upper fractal Weyl laws, of the Laplace-Beltrami operator on hyperbolic surfaces got some attention in recent decades motivated through application possibilities in physics. For Schottky surfaces, using the Selberg-Zeta function and transfer operator techniques, Guillopè-Lin-Zworski have established an upper fractal bound which is in line with a bound conjectured by Sjöstrand. For surfaces with finite volume and cusp, i.e. SL(2,Z), the distribution of resonances is also known. An upper bound in the case of infinite area surfaces with cusp (Hecke triangle surface) and unitary twist has been established by Naud-Pohl-Soares but the precise asymptotics of the resonance counting function aren't known yet.

In this talk, after a general introduction to resonances, the possible application of a different transfer operator technique will be investigated in order to obtain additional information about the asymptotics for the counting function in the aforementioned case of Hecke triangle surfaces while using a non-unitary twist.