Random groups have fixed points on CAT(0) cube complexes

A new construction of CAT(0) cube complexes

We introduce the notion of coupled link cube complex (CLCC) as a means of constructing interesting cocompactly cubulated groups. CLCCs are defined locally, making them a useful tool when precise control over the links is required. In this paper we study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria we produce a number of cubulated 3 and 4 manifolds with hyperbolic fundamental group.

Cube-like Incidence Complexes and Their Groups

The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many structural properties in common with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes have repeatedly appeared in applications.

Ping pong on CAT(0) cube complexes

Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. We show that for any finite collection of simultaneously inessential subgroups $\{H_1, \ldots, H_k\}$ in $G$, there exists an element $g$ of infinite order such that $\forall i$, $\langle H_i, g\rangle \cong H_i * \langle g\rangle$. We apply this to show that any group, acting faithfully and geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has property $P_{naive}$ i.e. given any finite list $\{g_1, \ldots, g_k\}…

Cube-Like Polytopes and Complexes

The main purpose of this paper is to popularize Danzer's power complex construction and establish some new results about covering maps between two power complexes. Power complexes are cube-like combinatorial structures that share many structural properties with higher-dimensional cubes and cubical tessellations on manifolds. Power complexes that are also abstract polytopes have repeatedly appeared somewhat unexpectedly in various contexts, although often under a different name. However, the non-polytope case is largely unexplored.

The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the…

立方体复形的同伦等价边界

Homotopy equivalent boundaries of cube complexes

本文研究有限维 CAT(0) 立方体复形的多种边界，通过 Roller 边界商上的偏序构造单纯 Roller 边界，并证明其与 Tits、单纯边界在同伦意义下 $Aut(X)$-等变等价，从而说明度量扰动不改变等变同伦类型。

CAT(0)立方复形中的循环双曲性

Cyclic hyperbolicity in CAT(0) cube complexes

本文证明余紧特殊群不含F2×F2当且仅当循环双曲，并在CAT(0)立方复形几何作用群中给出结构定理和强Tits替代，推论SQ-普遍性。

$\ell^p$ metrics on cell complexes

Motivated by the observation that groups can be effectively studied using metric spaces modelled on $\ell^1$, $\ell^2$, and $\ell^\infty$ geometry, we consider cell complexes equipped with an $\ell^p$ metric for arbitrary $p$. Under weak conditions that can be checked locally, we establish nonpositive curvature properties of these complexes, such as Busemann-convexity and strong bolicity. We also provide detailed information on the geodesics of these metrics in the special case of CAT(0) cube complexes.

Projection complexes and quasimedian maps

We use the projection complex machinery of Bestvina--Bromberg--Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the group has a BBF colouring and its associated hyperbolic spaces are quasiisometric to trees, then the group is quasiisometric to a finite-dimensional CAT(0) cube complex. We deduce various properties, including the Helly property for hierarchically quasiconvex subsets.

关于CAT(0)立方复形嵌入到树积中的研究

On embeddings of CAT(0) cube complexes into products of trees

本文证明了二维CAT(0)立方复形的最大度数Δ的接触图可被最多MΔ^26种颜色着色，从而证明该复形可嵌入到有限个树的积中，同时给出了无法嵌入到有限树积中的五维立方复形的例子。

The collapsibility of some CAT(0) simplicial complexes of dimension 3

We study the collapsibility of finite simplicial complexes of dimension 3 endowed with a CAT(0) metric. Our main result states that, under an additional hypothesis, finite simplicial 3-complexes endowed with a CAT(0) metric collapse to a point through CAT(0) subspaces.

CAT(0)偶数边形2-复形的平行墙定理

The Parallel Wall Theorem for CAT(0) even 2-complexes

本文证明了由偶数边形正多边形构造的CAT(0) 2-复形的平行墙定理，扩展了Janzen-Wise和Hruska-Wise的先前工作。

Superrigidity of actions on finite rank median spaces

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. In previous work, we introduced "Rol…

Examples of $2$-unrectifiable normal currents

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ normal currents can be realized as limits in the flat distance of currents associated to cube complexes.

CAT(0) 空间中次线性 Morse 测地线的帘幕刻画

Curtain Characterization of Sublinearly Morse Geodesics in CAT(0) Spaces

本文证明每个 CAT(0) 空间的次线性 Morse 边界可连续嵌入某双曲空间的 Gromov 边界，利用帘幕机制给出多种刻画，并回答了 Petyt-Spriano-Zalloum 的多个问题。

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 …

A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes

This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Miller, Owen and Provan's algorithm (2015) as a subroutine. Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.