The curve complex of the 3-holed projective plane admits an exhaustion by finite rigid sets, its simplicial automorphism group is isomorphic to the mapping class group, and it is quasi-isometric to a simplicial tree.
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A note on the curve complex of the 3-holed projective plane
The curve complex of the 3-holed projective plane admits an exhaustion by finite rigid sets, its simplicial automorphism group is isomorphic to the mapping class group, and it is quasi-isometric to a simplicial tree.