The mapping class group and it’s 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 the stabilizer of a framing of the surface. These are subgroups that Calderon and Salter have shown are important in the algebraic geometry of Moduli space. This 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.