相关论文: Finite blocking property versus pure periodicity
For any infinite-type surface $S$, a natural question is whether the homology of its mapping class group contains any non-trivial classes that are supported on (i) a compact subsurface or (ii) a finite-type subsurface. Our purpose here is…
In this paper, we consider inverse limits of [0,1] using upper semicontinuous set-valued bonding functions with the intermediate value property. Expanding on classical results by Barge and Martin, we explore the relationship between…
Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$,…
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
We show that on any Riemann surface S of genus g>1 any nonsingular even spin bundle defines e-foloation of S. When a surface is hyperelliptic then all leaves of this foliation are finite and almost all of them consists of 2g+2 points.…
We prove the existence of nonperiodic, properly embedded minimal surfaces in $\mathbb{R}^2\times\mathbb{S}^1$ with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).
We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the…
A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly…
A \emph{surface of translation} is a sum $(u,v)\mapsto\gt\alpha(u)+\gt\beta(v)$ of two space curves: a \emph{path} $\gt\alpha$ and a \emph{profile} $\gt\beta$. A fundamental problem of differential geometry and shell theory is to determine…
By a result of Blokh from 1984, every transitive map of a tree has the relative specification property, and so it has finite decomposition ideal, positive entropy and dense periodic points. In this paper we construct a transitive dendrite…
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which…
A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for…
Consider a finite morphism f:X -> Y of smooth projective varieties over a finite field k. Suppose X is the vanishing locus in projective N-space of at most r forms of degree at most d. We show there is a constant C, depending only on N, r,…
We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmuller orbits are recurrent to a compact subset of $SL(2;R)/SL(S)$, where $SL(S)$ is the Veech group of the surface. In this…
We prove that, for any infinite-type surface $S$, the integral homology of the closure of the compactly-supported mapping class group $\overline{\mathrm{PMap}_c(S)}$ and of the Torelli group $\mathcal{T}(S)$ is uncountable in every positive…
We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are…
The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In…
In this article, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder $M\simeq S^1\times\mathbb{R}$ or a complete Riemannian plane $M\simeq\mathbb{R}^2$ leads to having…
We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a…
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat…