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.
Finite Rigid Sets in Curve Complexes of Non-Orientable Surfaces
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abstract
A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus $g$ with $n$ holes for $g+n \neq 4$.
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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.