Abstract. In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to get a boundary for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.

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

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasiisometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.

We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher.

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$-injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (2) $u(\epsilon) = \chi(\Theta_u)$, (3) $\Theta_u \subset X$ is a section of the semiflow and the first return map to $\Theta_u$ is an expanding irreducible train track map representing $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.Comment: v7: Minor organizational and stylistic changes incorporating referee's suggestions. Notably, section 6.3 in v6 has been moved to section 4.5 in v7. 67 pages, 13 figures. Final version; accepted for publication in Geometry & Topolog

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity.

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

The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms φ : S → S, ranging over all surfaces S. More precisely, we consider pseudo-Anosovs φ : S → S with |χ(S)| log(λ(φ)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resulting set of mapping tori is finite. Said differently, there is a finite set of fibered hyperbolic 3-manifolds so that all small dilatation pseudo-Anosovs occur as the monodromy of a Dehn filling on one of the 3-manifolds in the finite list, where the filling is on the boundary slope of a fiber. * The authors gratefully acknowledge support from the National Science Foundation.Penner proved that there exists constants 0 < c 0 < c 1 so that for all closed surfaces S with χ(S) < 0, one has c 0 ≤ L(S)|χ(S)| ≤ c 1 .The proof of the lower bound comes from a spectral estimate for Perron-Frobenius matrices, with c 0 > log(2)/6 (see [Pe] and [Mc2]). As such, this lower bound is valid for all surfaces S with χ(S) < 0, including punctured surfaces. The upper bound is proven by constructing pseudo-Anosov homeomorphisms φ g : S g → S g on each closed surface of genus g ≥ 2 so that λ(φ g ) ≤ e c 1 /(2g−2) ; see also [Ba]. The best known upper bound for {L(S g )|χ(S g )|} is due to Hironaka-Kin [HK] and independently Minakawa [Mk], and is 2 log(2 + √ 3). The situation for punctured surfaces is more mysterious. There is a constant c 1 so that the upper bound of (1) holds for punctured spheres and punctured tori; see [HK, Ve, Ts]. However, Tsai has shown that for a surface S g,p of fixed genus g ≥ 2 and variable number of punctures p, there is no upper bound c 1 for L(S g,p )|χ(S g,p )|, and in fact, this number grows like log(p) as p tends to infinity; see [Ts].One construction for small dilatation pseudo-Anosov homeomorphisms is due to Mc-Mullen [Mc2]. The construction, described in the next section, uses 3-manifolds and is the motivation for our results.