Related papers: Train track complex of once-punctured torus and 4-…

Let $g, n \geq 0$ and $\Sigma = \Sigma_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $\Sigma$ has a rigid relative automorphism group, whose elements fix each monodromies…

In this paper we completely classify symplectic actions of a torus $T$ on a compact connected symplectic manifold $(M, \sigma)$ when some, hence every, principal orbit is a coisotropic submanifold of $(M, \sigma)$. That is, we construct an…

We define train track maps for graphs-of-groups $\cal G$ and exhibit the precise conditions under which the fundamental finiteness properties known for classical train track maps extend to this generalization. These finiteness properties…

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn…

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some…

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.

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…

Various curve complexes with vertices representing multicurves on a surface $S$ have been defined, for example [3], [4] and [8]. The homology curve complex $\mathcal{HC}(S,\alpha)$ defined in [7] is one such complex, with vertices…

For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to…

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

For a closed symplectic manifold $(M,\omega)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $\gamma$ we define a Floer complex based on 1-periodic…

For a compact oriented smooth $n$-manifold $M$ and a codimension-$1$ homology class $\phi \in \operatorname{H}_{n-1}(M, \partial M)$, we investigate a simplicial complex $\mathcal{S}^\dagger(M, \phi)$ relating the properly embedded…

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group $G$ is said to be recognizable by spectrum, if every finite group isospectral with $G$…

We classify the cohomology classes of Lagrangian 4-planes $\P^4$ in a smooth manifold $X$ deformation equivalent to a Hilbert scheme of 4 points on a $K3$ surface, up to the monodromy action. Classically, the cone of effective curves on a…

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries.…

A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and…

A graph $G$ embedded in a surface $S$ is called an $S$-grid when every facial boundary walk has length four, that is, the topological dual graph of $G$ in $S$ is 4-regular. Aside from the case where $S$ is the torus or Klein bottle, an…

Suppose that S is a surface of genus two or more, with exactly one boundary component. Then the curve complex of S has one end.

We completely classify the orientable infinite-type surfaces $S$ such that $\operatorname{PMap}(S)$, the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of…

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. In a previous paper, we study the case when the fixed point set…