Existence of strict transfer operator approaches for non-compact developable hyperbolic orbisurfaces

Abstract

For geometrically finite non-compact hyperbolic orbisurfaces fulfilling mild assumptions, we provide transfer operator families whose Fredholm determinant functions are identical to the respective Selberg zeta function. Our proof yields an algorithmic and uniform construction. By application of the cusp expansion algorithm by Pohl and introduction of a similar algorithmic procedure for orbisurfaces without cusps, we establish cross sections for the geodesic flow on the considered orbisurfaces, that yield highly faithful, but, in general, non-uniformly expanding discrete dynamical systems modeling the geodesic flow. The central object for this pursuit is the set of branches, which encapsulates the structure guaranteeing the cross section to be suitable for its purpose. These sets of branches are introduced and extensively studied. Through a number of algorithmic steps of reduction, elimination, and acceleration on a set of branches, we turn the associated cross section into one that yields a still highly faithful, but now uniformly expanding discrete dynamical system. By virtue of the strict transfer operator approach in the sense of Fedosova and Pohl, this gives rise to a family of transfer operators nuclear of order zero on a well-chosen Banach space, and the Fredholm determinant function is seen to admit a meromorphic continuation to the whole complex plane and to equal the Selberg zeta function. All statements allow for the inclusion of finite-dimensional representations with non-expanding cusp monodromy, in the sense that a twisted version of the Selberg zeta function as well as twisted transfer operators may be considered. A comprehensive overview of the required background knowledge in hyperbolic geometry precedes the investigations.