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 c and radius r r , we define the sphere of radius r r to be the induced subgraph on the set of vertices of distance r r from c c . We show that these spheres are always connected in high enough complexity, and prove a slightly weaker result for low complexity surfaces.
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Alex Wright (2024) studied this question.
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