**Tufts Meeting**

Saturday, December 9, 9AM-12:15PM

Bromfield-Pearson, Room 101

Talks:

Title: Covers, curve complexes, and hyperbolic 3-manifolds

Abstract: Let X be a closed orientable surface of genus at least 2. The curve complex of X is a simplicial complex that encodes the combinatorics of how simple closed curves intersect on X. In this talk, we'll introduce the curve complex and discuss its connections to hyperbolic 3-manifolds and to Teichmuller theory. The goal will be to state a result relating covering spaces of X to the geometry of its curve complex, and to give a rough idea of how the proof uses hyperbolic geometry. This represents joint work with Priyam Patel and Sam Taylor.

Title: Boundaries of hyperbolic and CAT(0) groups

Abstract: In the land of hyperbolic groups, there are important connections between algebraic properties of the group and topological properties of the visual boundary. We will discuss some of these result, in particular a major result which says hyperbolic groups which do not split over finite groups have connected and locally connected boundaries, and hence path connected boundaries. For CAT(0) groups, the connections are less clear. We will address some of the questions we might want to ask about CAT(0) groups and their boundaries and partially answer these questions.

Title: Dynamics on M_2: square-tiled surfaces, Teichmuller curves and modular curves.

Abstract: In my talk I will address a basic question about topological branched covers. Call two finite covers over a torus branched over a single point of the same type, if there is an orientation preserving homeomorphism of the torus that lifts to the homeomorphism of the covers. How many types of covers of fixed genus and degree are there? The answer to this naive question is not known in the simplest case of genus 2.

Closer investigation of this question will bring us to the notions of square-tiled surfaces and Teichmuller curves. I will also define a flat conformal metric on the modular curve X(d), in which it decomposes into unit squares identified along parallel sides, and show how does it help to answer the main question. In particular, we will see “pagoda" structures of X(d) that arise in the case of prime d.