Related papers: The end of the curve complex

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

The authors prove that for a closed surface of genus at least 3, the graph of pants decompositions has only one end.

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 from X…

We analyze the number of ends of the mapping class group of a stable avenue surface. We prove that the mapping class group is one-ended whenever the stable avenue surface has at least one end of discrete type. Our method is to show that the…

For each closed surface of genus $g\ge3$, we find a finite subcomplex of the separating curve complex that is rigid with respect to incidence-preserving maps.

It is proved that the curve graph $C^1(\Sigma)$ of a surface $\Sigma_{g,n}$ has a local pathology that had not been identified as such: there are vertices $\alpha,\beta$ in $C^1(\Sigma)$ such that $\beta$ is a dead end of every geodesic…

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.

We prove that the non-separating curve complex of every surface of finite type and genus at least three admits an exhaustion by finite rigid sets.

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the…

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…

Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.

Given a compact, oriented surface $S$ of finite genus and finitely many boundary components, we provide examples of finite covers $\tilde{S}$ of $S$ and non-simple closed curves $\gamma$ on $S$ which lifts to simple closed curves on…

We show that the complex of homologous curves of a closed, oriented surface of genus g is (g-3)--acyclic.

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the…

We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…

We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of…