Automorphisms of curve complexes on nonorientable surfaces

Abstract

Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group MN of N .

Finite rigid sets in curve complexes of nonorientable surfaces

“…The mapping class group of the surface acts on it by simplicial automorphisms in a natural way, inducing an isomorphism, except for a few sporadic cases, from the mapping class group onto the group of simplicial automorphisms of the curve complex. These results are proved by Ivanov [12], Korkmaz [13] and Luo [14] in the orientable case, and by Atalan-Korkmaz [3] in the non-orientable case.…”

“…A top dimensional pant decomposition contains exactly g essential one-sided curves (cf. the proof of Lemma 2 above given in [3]). By Lemma 2, we have the following corollary.…”

“…The pair of pants X 2 glued to X 1 must be of type (2); otherwise, i.e., if type (3), the surface would be N 1,3 , which is impossible. The pair of pant X 3 glued to X 1 ∪ X 2 can be of type (2) or (3). If type (3), then N = N 1,4 .…”

“…For non-orientable surfaces, the curve complex was first studied by Scharlemann [17]. Atalan-Korkmaz [3] proved that the natural map from the mapping class group Mod(N ) of a connected non-orientable surface N of genus g with n holes to the automorphism group Aut(C(N )) of the curve complex of N is an isomorphism for g +n ≥ 5, Irmak generalized this result by showing that superinjective simplicial maps and injective simplicial maps C(N ) → C(N ) are induced by homeomorphisms N → N (cf. [10] and [11]).…”

“…[3, Proposition 2.3] Let N be a connected non-orientable surface of genus g ≥ 2 with n holes. Suppose that (g, n) = (2, 0).…”