The admissible curve graph is not hyperbolic

Jacob Russell (Swathmore)

Abstract: The mapping class group and its subgroups are often illuminated by actions on graphs built from curves on the surface. These actions allow for a variety of questions about the group to be translated into either combinatorial or geometric information about these graphs. We will examine this approach in the case of a stabilizer of a vector field on the surface. These are subgroups that Calderon and Salter have shown are important for the algebraic geometry of Moduli space. Their work also suggests that the appropriate graph for these subgroups to act on is the graph of curves with winding number zero. We show the geometry of this graph can be well understood using Masur and Minsky's subsurface projections. As a consequence, we learn that, unlike the traditional curve graph, this admissible curve graph is not hyperbolic.