Related papers: Exhausting curve complexes by finite rigid sets

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.

We prove that for a convex polytope decomposition of a domain in $\mathbb R^n$, an integral curve of a piecewise analytic vector field is chopped by the decomposition into finitely many pieces. As a consequence, we prove the finiteness of…

We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike…

Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while…

We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2-dimensional…

Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes…

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate…

The purpose of this article is to investigate a hyperbolic complex manifold $M$ exhausted by a pseudoconvex domain $\Omega$ in $\mathbb C^n$ via an exhausting sequence $\{f_j\colon \Omega\to M\}$ such that $f_j^{-1}(a)$ converges to a…

We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We…

We proved that the union of rational curves is dense on a very general K3 surface and the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology.

Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature…

We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

A Coxeter group acts properly and cocompactly by isometries on the Davis complex for the group; we call the quotient of the Davis complex under this action the Davis orbicomplex for the group. We prove the set of finite covers of the Davis…

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map…

Using pseudoholomorphic curves techniques from symplectic geometry, Barney Bramham proved recently that every smooth irrational pseudo-rotation of the unit disk is the limit, for the $C^0$ topology, of a sequence of smooth periodic…

The combination of this paper and its companion complete the classification of monodromy groups of indecomposable coverings of complex curves $f:X\rightarrow \mathbb P^1$ of sufficiently large degree in comparison to the genus of $X$. In…