Asymptotically geodesic surfaces

Let M be a hyperbolic 3-manifold. We say a sequence of distinct (non-commensurable) essential closed surfaces in M is asymptotically geodesic if their principal curvatures go uniformly to zero. When M is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they do not always equidistribute. We will also talk about the fact that such sequences do not exist when M is geometrically finite of infinite volume. Finally, time permitting, we will discuss some partial answers to the question: does the existence of asymptotically geodesic surfaces in a negatively-curved 3-manifold imply the manifold is hyperbolic? This joint work with Ben Lowe.