Today I'm going to talk about a startling correspondence between *hyperbolic surfaces* [write], which are negatively curved, and *half-translation surfaces* [write], which are a special kind of flat surface. I think this correspondence deserves to be more widely known: you can find pieces of it sort of buried in the works of Bonahon, Casson and Bleiler, and others, but I've never seen the whole picture laid out in one place, so I'm hoping to lay out a good part of it here. hyperbolic surfaces | half-translation surfaces ============ These two kinds of surfaces [point] are the stars of the show, so let's start by getting to know them better. ------------ A *hyperbolic surface* is a surface patched together from pieces of the hyperbolic plane, with the pieces identified using isometries. It inherits all the local geometric features of the hyperbolic plane, including a way of measuring distances and a way of recognizing "straight lines," or *geodesics*. The universal cover of a hyperbolic surface is isometric to the hyperbolic plane itself. Today we'll be hearing about compact hyperbolic surfaces, but we'll be looking at their universal covers, viewed through the fisheye lens of the Poincaré disk model. ------------ A *half-translation surface* is a surface patched together from pieces of the Euclidean plane, using two kinds of identifications: translations, which you're familiar with, and moves I'll call *flips*, which are translations composed with 180° rotations. A half-translation surface inherits all the geometric features of the plane which are invariant under rotations and flips. These include distances and geodesics, but also a lot of other stuff. For example, you can foliate the Euclidean plane by horizontal lines, and that foliation is invariant under rotations and flips. As a result, every half-translation surface comes with a global foliation, called the *horizontal foliation*, which is a useful way of visualizing the half-translation structure. If you care to deal only with honest half-translation surfaces, and you want them to be compact too, you're going to find yourself very lonely: it'll be just you and the torus. It's more fun to allow half-translation surfaces to have a certain kinds of singularities. The simplest one, and the only one we really need today, is a conical singularity made from three half-planes stitched together around a single point. The three leaves of the horizontal foliation that run into the singularity are called *critical leaves*. ============ Now that we know what hyperbolic and half-translation surfaces are, I'd like to describe some of their geometric and dynamical features, both to emphasize how different they are and to point out a few similarities. ------------ Hyperbolic surfaces are *negatively curved*: if you take a closed loop, like this triangle, and carry a tangent vector around it counterclockwise, the vector will come back rotated clockwise from its original direction. In fact, hyperbolic surfaces are *uniformly* negatively curved: the change in angle of the vector is proportional to the area of the loop. This triangle has area π/3, so the vector got rotated by -π/3. With a stretch of the imagination, you can even carry a tangent vector around an *ideal triangle*, whose corners lie on the mythical "boundary at infinity" of the universal cover. As you carry the vector off toward infinity, it whirls around the surface faster and faster, and then you do a little sleight of hand and suddenly the vector is backing up along the next edge of the triangle. When you get back to where you started, the vector is exactly backward from its original direction. That's no surprise, because an ideal triangle has area π, so it encloses a total curvature of -π. ------------ Half-translation surfaces are modeled on the flat Euclidean plane, so they have no curvature---except at the singularities. If you carry a tangent vector clockwise around a singularity, once again, it ends up backward. A loop around a singularity of a half-translation surface, like an ideal triangle on a hyperbolic surface, encloses a total curvature of -π, but that curvature is concentrated at a single point. hyperbolic surfaces | half-translation surfaces ______________________________________________________________________ curvature -π in each ideal triangle | curvature -π at each singularity ------------ Another big difference between hyperbolic and half-translation surfaces comes from the dynamics of things moving in straight lines. I mentioned earlier that every half-translation surface comes with a horizontal foliation, which comes from the foliation of the plane by horizontal lines. The leaves of the horizontal foliation are non-intersecting geodesics, so we might call this foliation a *geodesic foliation*. A half-translation surface has an RP^1's worth of geodesic foliations---one for every family of parallel lines in the plane. Notice the critical leaves spinning around the singularities as I change the slope. At some slopes, the critical leaves run into each other, creating features called *saddle connections*. hyperbolic surfaces | half-translation surfaces ______________________________________________________________________ curvature -π in each ideal triangle | curvature -π at each singularity | | can be foliated by geodesics A compact hyperbolic surface can't be foliated by non-intersecting geodesics---it's just too hard to keep the geodesics from running into each other. In fact, it's hard enough just to keep geodesics from running into themselves. Birman and Series showed in the '80s that the full set of non-self-intersecting geodesics on a compact hyperbolic surface only covers a region of Hausdorff dimension one---nowhere near the whole surface ["Geodesics with bounded intersection numbers on surfaces are sparsely distributed," 1985 | via Bonahon, "Geodesic laminations on surfaces," 1997]. If we can't cover the whole surface with non-intersecting geodesics, we might as well see how much if it we can cover. For example, we can look for closed subsets of the surface foliated by non-intersecting geodesics, and then keep growing them until they can't get any bigger. We end up with these intricate patterns, called *maximal geodesic laminations*. The geodesics that make up the lamination are called its *leaves*. Maximal godesic laminations are really weird. A maximal geodesic lamination always has uncountably many leaves, but its complement is a finite collection of ideal triangles, which you can see clearly on the universal cover. There are two kinds of leaves: the leaves on the boundaries of the triangles, which lift to a countable set of geodesics on the universal cover, and the other leaves, which are uncountable. To make these things feel a little more familiar, it's useful to look at an arc transverse to the lamination. The intersection of the arc and the lamination is homeomorphic to a Cantor set, plus maybe some isolated points corresponding to any isolated leaves. hyperbolic surfaces | half-translation surfaces _____________________________________________________________________ curvature -π in each ideal triangle | curvature -π at each singularity | can't be foliated by geodesics, | can be foliated by geodesics but can carry maximal geodesic | laminations |