Two of the most studied invariants of a closed hyperbolic surface are the Laplace and the Length spectrum. Even though a complete description of these spectra is difficult to obtain in general, one can rely on good asymptotics for counting the number of eigenvalues of the Laplacian (respectively, length of closed geodesics) in a certain window, as this window expands to infinity.
Changing the perspective, one can fix an interval and ask for asymptotics on the number of eigenvalues (respectively, length of closed geodesics) in that interval as the volume grows to infinity. This bears the question on how to obtain sequences of surfaces with increasing volume which give good asymptotics. In this talk I will introduce two such notions, namely Benjamini-Schramm and Plancherel convergence, and discuss their relation.
This talk is based on a joint work with C. Kamp.