The main result of this paper is that every Riemann surface has a Hall ray at each cusp. By this we mean that the spectrum of maximal penetration heights of geodesics into a horocycle about the cusp fills out a real half-line. For the modular surface, this result is well known and derives from Hall's Theorem for continued fractions.

We also show that a Hall ray can exist without the presence of cusps in two settings: First, on a surface derived as a limit of cusped surfaces, whose fundamental region contains two entire horocycles. And second, with respect to a hyperbolic continued fraction for which the former role of a cusp is played by a simple closed geodesic.

The limiting process mentioned above also produces an infinite class of closed geodesics on the theta surface, the quotient of the upper half-plane by the usual theta group, that are pair-wise equal in length, but whose precursors in the limit process are never equalm—this equality is then accidental. That is, there is a change in length spectrum multiplicity at the limit surface.

We consider nonnegatively curved $4$-manifolds that admit effective isometric actions by finite groups and from that draw topological conclusions about the manifold. Our first theorem shows that if the manifolds admits an isometric $Z_{p} \times Z_{p}$, for $p$ large enough that the manifold has Euler characteristic less than or equal to five. Our second theorem requires no hypothesis on the structure of the group other then that it be large but it does require the manifold to be $\delta$-pinched, in which case we can then again conclude that the Euler characteristic is less than or equal to five.