We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map X → C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) point-wise stabilizer of X in Aut(C(S)). Furthermore, if S is not a twice-punctured torus, then we can replace Aut(C(S)) in this statement with the extended mapping class group Mod ± (S). arXiv:1206.3114v3 [math.GT] 24 Jul 2012Proof. We first observe that if γ = σ J k ∈ P \{σ J 1 }, then σ J k , α i+k+1 are Farey neighbors in the four-holed sphere N (σ J k ∪ α i+k+1 ), with boundary componentsFor γ = β + J k , we again let γ = α i+k+1 and consider the pants decompositions Figure 9. Using these pants decompositions, we see thatJ k is proven in exactly the same way as β + J k , and this completes the proof of the claim. As in Section 4, it follows that, for all i, N (φ(α i ) ∪ φ(α i+1 )) has only one boundary component in S. Since S has no punctures, N (φ(α i ) ∪ φ(α i+1 )) is a one-holed torus and thus i(φ(α i ), φ(α i+1 )) = 1.Since S has genus g, we deduce that there is a homeomorphism h 0 : S → S satisfying h 0 (α) = φ(α) for all α ∈ C. Moreover, h 0 is unique up to precomposing with elements in the point-wise stabilizer H C of C.Next, each curve σ J ∈ S is uniquely determined by the almost filling set α J ∪α J ⊂ C, where J = {i, . . . , i + k} and J = {i + k + 2, . . . , i − 2} (recall the indices are taken modulo 2g + 2). Since φ(σ J ) is disjoint from all curves in φ(α J ) ∪ φ(α J ), then φ(σ J ) is also uniquely determined by φ(α J ) ∪ φ(α J ), and thus φ(σ J ) = h 0 (σ J ), for all σ J ∈ S J .An analogous argument almost works for a bounding pair β ± J ∈ B. Here, {β + J , β − J } is similarly determined by α J ∪ α J . As in the previous paragraph, we can concludeHowever, it is not necessarily the case that φ(β ± J ) = h 0 (β ± J ), since H C acts nontrivially on B. We therefore wish to precompose h 0 with some element f ∈ H C so that h 0 • f (β ± J ) = φ(β ± J ). To choose the appropriate element f ∈ H C we proceed as follows. The φimages of the even (respectively, odd) chain curves cut S into two complementary surfaces, each homeomorphic to S 0,g+1 , which we denote Ω ± Now, a bounding pair β i,j,+ J
It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.
We prove that every simplicial automorphism of the free splitting graph of a free group F n is induced by an outer automorphism of F n for n ≥ 3.
Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the same hypotheses on genus, if $X$ and $Y$ have finite analytic type then every non-constant holomorphic map $\CM(X)\to\CM(Y)$ between the corresponding moduli spaces is a forgetful map. In particular, there are no such holomorphic maps unless $X$ and $Y$ have the same genus and $Y$ has at most as many marked points as $X$
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multitwists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary. 57M07, 20F34; 57M60, 30F60, 32G15, 57R50
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