Calendar

On a surface, the curve graph, defined by Harvey, is a 1-complex whose vertices are isotopy class of simple closed curves and edges correspond to disjointness. Masur and Minsky famously proved that this graph is Gromov hyperbolic. Here we will examine a family of similar graphs, defined by Hamenstädt, where edges are determined by a complexity condition on the two curves. More precisely rather than just asking if the two curves intersect we want to measure the complexity of the intersection. When the two curves “fill’’ the surface (every curve intersects one of the two) then complementary regions will be a collection of even sided polygons. The complexity is highest when these complementary polygons are all hexagons and quadrilaterals and in the principal curve graph there is an edge between two curves whenever the two curves intersection is not of this maximal complexity. By changing the complexity threshold we get a sequence of graphs (and maps) that interpolate between the original curve graph and the principal graph. We show that principal curve graph is a quasi-tree (a strong hyperbolicity condition) and, more generally, for any of the graphs in the sequence the pre-image of a bounded set in one graph is a quasi-tree in the graph one level up. This is joint work with Mladen Bestvina and Alex Rasmussen.