Arc complexes, sphere complexes and Goeritz groups

Finite Rigid Sets in Arc Complexes

For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by fin…

Combinatorial rigidity of arc complexes

We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. As an application we deduce some rigidity results for the Fomin-Shapiro-Thurston cluster algebra associated to such a surface. Our proofs do not employ any known simplicial rigidity result.

细曲线和细弧复形的同调类型

Homotopy types of fine curve and fine arc complexes

本文证明了细曲线复形与曲线复形同调等价，并证明细弧复形可收缩。

Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Go…

部分装饰双曲多边形的弧复形

The arc complexes of partially decorated hyperbolic polygons

该研究针对部分装饰的理想与理想一次穿孔双曲多边形，在组合框架下证明其弧复形为闭分段线性球，并借此实现这类多边形弱加长形变的参数化。

The Structure and Singularities of Arc Complexes

A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The main result of this paper is the determination of those arc complexes Arc(F) that are also spherical. This classification has consequences for Riemann's moduli space via its known identification with an analogous arc complex in the punctured case with no boundary. Namely, the esse…

Matching arc complexes: connectedness and hyperbolicity

Addressing a question of Zaremsky, we give conditions on a finite simplicial graph which guarantee that the associated matching arc complex is connected and hyperbolic.

Contractible, hyperbolic but non-CAT(0) complexes

We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of …

Plant complexes and homological stability for Hurwitz spaces

We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations. In the first part, we introduce and study plant complexes, a large new class of simplicial complexes, generalizing the arc complex on a surface with marked points. In the second part, we generalize a result by Ellenberg-Venkatesh-Westerland by showing that homological stabilization…

单纯复形的链条件与 CUB 空间

A link condition for simplicial complexes, and CUB spaces

本文研究了具有唯一凸测地双组合的 CUB 空间，并提出了单纯复形成为 CUB 空间的链条件，推广了 Gromov 链条件，适用于欧氏建筑、Artin 复形等多种几何结构。

格、Garside结构与弱模图

Lattices, Garside structures and weakly modular graphs

本文研究了具有 $\widetilde A_n$ 形单纯形的单纯复形的组合非正曲率性质，证明了具有保序 $\mathbb Z$ 作用的格及其格商均构成弱模图，并揭示了范畴 Garside 结构与格的关系，给出了具有奇异性质的 Garside 群首批实例。

The hyperbolicity of the sphere complex via surgery paths

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex. We explain how to deduce from our proof the hyperbolicity of some other comple…