Spheres in the curve graph and linear connectivity of the Gromov boundary

Abstract

We consider the curve graph in the cases where it is not a Farey graph, and show that its Gromov boundary is linearly connected. For a fixed center point c and radius r, we define the sphere of radius r to be the induced subgraph on the set of vertices of distance r from c. We show that these spheres are always connected in high enough complexity, and prove a slightly weaker result for low complexity surfaces.