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Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(N)$ be the curve complex of $N$. We prove that if $(g,n) = (3,0)$ or $g + n \geq 5$, then there is an exhaustion of $\mathcal{C}(N)$ by a sequence of finite rigid sets. This improves the author's result on exhaustion of $\mathcal{C}(N)$ by a sequence of finite superrigid sets., Comment: 19 pages, 13 figures. arXiv admin note: text overlap with arXiv:1903.04926

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(N)$ be the curve complex of $N$. We prove that if $(g, n) \neq (1,2)$ and $g + n \neq 4$, then there is an exhaustion of $\mathcal{C}(N)$ by a sequence of finite superrigid sets., Comment: 28 pages, 15 figures. arXiv admin note: text overlap with arXiv:1707.09937

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(R)$ be the curve graph of $R$. We prove that if $g=0, n \geq 5$ or $g=1, n \geq 3$, and $\lambda : \mathcal{C}(R) \rightarrow\mathcal{C}(R)$ is an edge preserving map, then $\lambda$ is induced by a homeomorphism of $R$, and this homeomorphism is unique up to isotopy., Comment: 25 pages, 14 figures. arXiv admin note: text overlap with arXiv:1708.05290

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $\lambda : \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map, then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $\theta : \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then $\theta$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $\tau: \mathcal{HT}(R) \rightarrow\mathcal{HT}(R)$ is a rectangle preserving map, then $\tau$ is induced by a homeomorphism of $R$. We also prove that these homeomorphisms are unique up to isotopy when $(g, n) \neq (2, 0)$., Comment: 38 pages, 25 figures. Changed the statement of Theorem 1.3 and gave more explanation in some proofs

Let $N$ be a compact, connected, non-orientable surface of genus $\rho$ with $n$ boundary components, with $\rho \ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $\varphi: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $\varphi (g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $\lambda :\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(\alpha) = \lambda(\alpha)$ for every vertex $\alpha$ in $\mathcal{T}(N)$ where $H=[h]$., Comment: 51 pages, 36 figures

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $\lambda$ is induced by a homeomorphism of $N$., Comment: 15 pages, 6 Figures. The paper was shortened and reorganized

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\lambda$ be a simplicial map of the complex of curves, $\mathcal{C}(N)$, on $N$ which satisfies the following: $[a]$ and $[b]$ are connected by an edge in $\mathcal{C}(N)$ if and only if $\lambda([a])$ and $\lambda([b])$ are connected by an edge in $\mathcal{C}(N)$ for every pair of vertices $[a], [b]$ in $\mathcal{C}(N)$. We prove that $\lambda$ is induced by a homeomorphism of $N$ if $(g, n) \in \{(1, 0), (1, 1), (2, 0)$, $(2, 1), (3, 0)\}$ or $g + n \geq 5$. Our result implies that superinjective simplicial maps and automorphisms of $\mathcal{C}(N)$ are induced by homeomorphisms of $N$., Comment: 13 pages, 6 figures. The paper was shortened and reorganized

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $(g, n) \in \{(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\}$ or $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary components., Comment: In response to comments from the referee, the paper was shortened and reorganized. A minor mistake pointed out by the referee was also corrected

We prove that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center., Comment: 18 pages, 14 figures

In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center., Comment: 15 pages, 8 figures (reorganized, shortened)

Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Aut(HT(S)) is isomorphic to the extended mapping class group of S modulo its center., Comment: Some editorial changes are made, the list of references is expanded

Let $R$ be a compact, connected, orientable surface of genus $g$, $Mod_R^*$ be the extended mapping class group of $R$, $\mathcal{C}(R)$ be the complex of curves on $R$, and $\mathcal{N}(R)$ be the complex of nonseparating curves on $R$. We prove that if $g \geq 2$ and $R$ has at most $g-1$ boundary components, then a simplicial map $\lambda: \mathcal{N}(R) \to \mathcal{N}(R)$ is superinjective if and only if it is induced by a homeomorphism of $R$. We prove that if $g \geq 2$ and $R$ is not a closed surface of genus two then $Aut(\mathcal{N}(R))= Mod_R^*$, and if $R$ is a closed surface of genus two then $Aut(\mathcal{N}(R))= Mod_R ^* /\mathcal{C}(Mod_R^*)$. We also prove that if $g=2$ and $R$ has at most one boundary component, then a simplicial map $\lambda: \mathcal{C}(R) \to \mathcal{C}(R)$ is superinjective if and only if it is induced by a homeomorphism of $R$. As a corollary we prove some new results about injective homomorphisms from finite index subgroups to $Mod_R^*$. The last two results complete the author's previous results to connected orientable surfaces of genus at least two., Comment: 24 pages, 13 figures; The result about automorphism group of complex of nonseparating curves has been extended to compact, connected, orientable surfaces of genus at least two

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R., Comment: 37 pages, 19 figures; The proof for surfaces of genus two with at least two boundary components is added

In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical braid groups, mapping class groups, and Torelli groups., Comment: 21 pages, 0 figures

Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $\lambda: \mathcal{C}(S) \to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $\alpha$ and $\beta$ are two vertices in $\mathcal{C}(S)$ and $i(\alpha, \beta) \neq 0$, then $i(\lambda(\alpha), \lambda(\beta)) \neq 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mod_S^*$ and $f: K \to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism of $S$ and $f$ has a unique extension to an automorphism of $Mod_S^*$., Comment: 34 pages, 17 figures