Group actions on median spaces

Lamplighter groups, median spaces, and Hilbertian geometry

From any two median spaces $X,Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,H$ and two (equivariant) coarse embeddings into median spaces $X,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. As an application, we prove that $$α_1(G \wr H) \geq \min(α_1(G),α_1(H))/2 \text{ for all finitely generated groups $G,H$,}$$ where $α_1(\cdot)$ denotes the $\ell^1$-compression. As an ot…

The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes

We define a bounded cohomology class, called the {\em median class}, in the second bounded cohomology -- with appropriate coefficients --of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the {\em median class of an action} by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result …

Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen, and that under certain conditions a group action $G \curvearrowright X$ induces an acylindrical action $G \curvearrowright \mathcal{C}X$, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto. As an application, we exhibit an…

中位空间上自由传递作用的构造指南

A guide to constructing free transitive actions on median spaces

本研究引入ore结构的新概念，构造了大量可在中位空间、完备通用实树及实树乘积上自由传递作用的群，证明了存在连续统多个不同构的相关群，丰富了几何群论中群与中位空间作用的研究成果。

Cubical-like geometry of quasi-median graphs and applications to geometric group theory

The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the definition of hyperplanes from CAT(0) cube complexes, and we show that the geometry of a quasi-median graph essentially reduces to the combinatorics of its hyperplanes. In the second part, we exploit the specific structure of the hyperplanes to state combination results. The main idea is that if a group acts in a suitable way on a quasi-median graph so that…

Coarse median structures and homomorphisms from Kazhdan groups

We study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan's property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock-Drutu-Sapir on the mapping class groups of ori…

Special cube complexes revisited: a quasi-median generalisation

In this article, we generalise Haglund and Wise's theory of special cube complexes to groups acting on quasi-median graphs. More precisely, we define special actions on quasi-median graphs, and we show that a group which acts specially on a quasi-median graph with finitely many orbits of vertices must embed as a virtual retract into a graph product of finite extensions of clique-stabilisers. In the second part of the article, we apply the theory to fundamental groups of some graphs of groups called right-angled graphs of groups.

Coarse medians and Property A

We prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have property A.

Kazhdan and Haagerup properties from the median viewpoint

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger.

Embeddings into Thompson's groups from quasi-median geometry

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of Thompson's group $V$ (resp. $T$, $F$). As an application, we show that several braided diagram groups embeds into Thompson's group $V$, including Higman's groups $V_{n,r}$, groups of quasi-automorphisms $QV_{n,r,p}$, and generalised Houghton's groups $H_{n,p}$.

从中位数几何视角看Cremona群的综述

Survey on Cremona groups from a median geometric point of view

本文基于2023年Riverside研讨会讲义，介绍Cremona群在中位数图上作用的最新构造，旨在同时服务于几何群论学家和代数几何学家。

所有映射类群和右角群的一族粗中位数结构

A continuum of coarse medians for all mapping class groups and right-angled groups

本文证明非零星曲面映射类群及一大类右角群具有一族等变粗中位数结构，通过拟上循环修改层级双曲群结构，改进先前定理且不依赖余紧立方化。

相对的端数和准中位图

Relative numbers of ends and quasi-median graphs

本文基于准中位图的动作，研究了相对端数和共端数的特性，扩展了Sageev对共维度一子群的刻画。

Characterising actions on trees yielding non-trivial quasimorphisms

We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in a…

Median structures on asymptotic cones and homomorphisms into mapping class groups

The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, a…

短层级双曲群 I：不可数多个粗中值结构

Short hierarchically hyperbolic groups I: uncountably many coarse median structures

本文证明五穿孔球面映射类群及某些右角Artin群具有不可数多个粗中值结构，通过发展短层级双曲群理论并利用拟态射构造拟直线实现。

A four point characterisation for coarse median spaces

Coarse median spaces simultaneously generalise the classes of hyperbolic spaces and median algebras, and arise naturally in the study of the mapping class groups and many other contexts. One issue with their definition as originally conceived by Bowditch is the need to establish median approximations for all finite subsets of the space, an approach which allowed the definition of rank (a proxy for dimension) in terms of the dimensions of the approximating spaces. Here we provide a simplification of the definition in terms of a $4$-point condition analogous to the $4$-point condition defining h…

关于 CAT(0) 立方复形上真作用不存在拟等距刚性

No quasi-isometric rigidity for proper actions on CAT(0) cube complexes

本文构造了在中间图上具有真余紧作用的群，其拟等距群无真作用，回答了 Niblo 等人的问题，例子为拟等距平凡中心扩张。

CAT(0) 立方体复形上的中位数拟态射及其杯积

Median quasimorphisms on CAT(0) cube complexes and their cup products

本文将自由群 Brooks 拟态射杯积的消没定理推广至 CAT(0) 立方体复形上的中位数拟态射，并应用于树作用群与直角 Artin 群。

紧中值代数以唯一方式成为$μ$-边界

Compact Median Algebras are $μ$-Boundaries in a unique way

该研究探讨群对紧中值代数的作用，证明在适当条件下，配备作用群上生成概率测度$μ$的紧中值代数可唯一实现为Furstenberg意义下的$μ$-边界，还得到相关结构性结果。

Median sets of isometries in CAT(0) cube complexes and some applications

In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.

超平面复形的同伦类型与右角Artin群的应用

Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups

研究通过分析准中位图中超平面复形的同伦类型，证明了右角Artin群的准等距性与特定复形的同伦等价性。

On the arboreal structure of right-angled Artin groups

The present article continues the study of median groups initiated in [6, 9, 10]. Some classes of median groups are introduced and investigated with a stress upon the class of the so called A-groups which contains as remarkable subclasses the lattice ordered groups and the right-angled Artin groups. Some general results concerning A-groups are applied to a systematic study of the arboreal structure of right-angled Artin groups. Structure theorems for foldings, directions, quasidirections and centralizers are proved.

保持粗中位数的自同构

Coarse-median preserving automorphisms

本文引入保持粗中位数的自同构，刻画 RAAG/RACG 不动子群的几何性质，并将无限阶外自同构实现为有限秩中位数空间上的同位似。

粗粒度立方体刚性

Coarse cubical rigidity

研究显示，许多右角艺术和科克塞特群的所有紧致立方体划分在粗粒度上看起来相同，诱导相同的粗粒度中位结构。该研究为非超几何群提供了此类性质的第一个例子。

Group actions with commensurated subsets, wallings and cubings

We study commensurating actions of groups and the associated properties FW and PW, in connection with wallings, median graphs, CAT(0) cubings and multi-ended Schreier graphs.

从中位几何视角看 Lin 与 Shinder 的一个定理

On a Theorem by Lin and Shinder through the Lens of Median Geometry

本文针对 Lin 与 Shinder 关于秩大于2的 Cremona 群到整数群存在非平凡同态的定理，给出基于中位几何的替代性理解视角。

关于特殊群的自同构与分解

On automorphisms and splittings of special groups

研究特殊群的外自同构群，探讨其无限性的条件及分解与广义德尼特变换的关系。

算子值Haagerup不等式的重心框架

A Centroid Framework for Operator-Valued Haagerup Inequalities

本文提出左正则表示的重心块分解，证明有限生成群的算子值Haagerup不等式，对两类群得到双边或带特定误差的球块估计，完善相关不等式的下界与估计结果。

Largest hyperbolic actions and quasi-parabolic actions in groups

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have largest hyperbolic actions, …

Non-nesting actions of groups on real trees vs isometric actions

We study non-nesting actions on R-trees. We prove that some natural conditions describing how the group is generated, imply that such an action involves an isometric action on an R-tree. This can be applied to permutation groups, linear groups and some Polish groups.

Dyer群具有同行者消去性质

Dyer groups have the falsification by fellow-traveller property

本文证明Dyer群具有同行者消去性质，并指出其在统一处理Coxeter群和循环群图积中的作用。

Abstract group actions of locally compact groups on CAT(0) spaces

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.

Some Remarks on Groups with Action on Itself

We implement GAP functions about groups with action on itself and investigate some basic properties of small groups with action on itself of order $<32$.

线性群与有限群表示中群元素的无不动点作用

Fixed Point Free Actions of Group Elements in Linear Groups and Finite Group Representations

本文综述线性群及群表示中元素出现特征值1（即无不动点作用）相关问题的研究现状。

Topological full groups of line-like minimal group actions are amenable

We consider a finitely generated group acting minimally on a compact space by homeomorphsims, and assume that the Schreier graph of at least one orbit is quasi-isometric to a line. We show that the topological full group of such an action is amenable.

Tarski numbers of group actions

The Tarski number of an action of a group G on a set X is the minimal number of pieces in a paradoxical decomposition of it. For any k>3 we construct a faithful transitive action of a free group of rank k-1 with Tarski number k. Using similar techniques we construct a group action of a free group F with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of F are arbitrarily large.

Actions of automorphism groups of Lie groups

This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and quasi-invariant measures, the induced actions on the spaces of probability measures on the groups, and results concerning various issues in ergodic theory, topological dynamics, smooth dynamical systems, and probability theory on Lie groups.

A virtual geometric action of a braid group

We study a geometric action on a CAT(0) space of a finite index subgroup of the quotient group of the braid group on 4 strands by its center.

2-Dimensional Groups with Action : The Category of Crossed Module of Groups with Action

In this paper, we define the notion of crossed modules of groups with action and investigate related structures. Functions for computing of these structures have been written using the GAP computational discrete algebra programming language.

On some examples of group actions and group extensions

We study and relate certain actions and extensions involving 2-groups.

Hyperbolic projections and topological invariance of sublinearly Morse boundaries

We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse bou…

群作用的图与群在树上的作用

Graphs of group actions and group actions on trees

本文提出群作用图理论，用于系统构建具有指定局部行为的群作用，统一了群图和局部作用图等方法。

Non-nesting actions of Polish groups on pretrees

We study non-nesting actions of groups on real trees. We prove some fixed point theorems for such actions under the assumption that groups are Polish and have comeagre conjugacy classes.

Action dimensions of some simple complexes of groups

The action dimension of a discrete group $G$ is the minimum dimension of contractible manifold that admits a proper $G$-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.

非恰当的表面群作用

An Improper Surface Group Action

本文证明了Long和Reid在1999年提出的表面群在树乘积上的作用并非恰当。

非几乎阿贝尔不连续群作用与齐性空间上的真SL(2,R)作用

Non-virtually abelian discontinuous group actions vs. proper SL(2,R)-actions on homogeneous spaces

本文开发算法验证半单齐性空间上离散群作用的真性，并发现不承认真SL(2,R)作用的非几乎阿贝尔不连续群作用新例。

Paradoxical decomposition of Group Actions and Configuration

In this paper we study the notion of configuration for group actions. It is proved that some properties concerning configuration of groups can be extended for the case of group actions. The relationship between configuration and different types of well-known ping-pong lemma is also investigated.

On topological full groups of Z^d-actions

We give new examples of simple finitely generated groups arising from actions of free abelian groups on the Cantor sets. As particular examples, we discuss groups of interval exchange transformations, and a group naturally associated with the Penrose tilings. Many groups in this class are amenable.

有限生成群作用的定量顺从性

Quantitative Amenability for Actions of Finitely Generated Groups

本研究将有限生成群的等周剖面概念推广至群作用，通过测量轨道图有限子图边界定义作用的等周剖面，证明本质自由作用的等周剖面衰减等价于Zimmer意义下的作用顺从性，并建立了保测作用与群的等周剖面的关联。

$C^*$-simplicity for groups with non-elementary convergence group actions

We prove that a countable group with an effective minimal non-elementary convergence group action is a Powers group. More strongly we prove that it is a strongly Powers group and thus its non-trivial subnormal subgroups are $C^*$-simple.

On the bordism group for group actions on the torus

In this short note, we study the bordism problem for group actions on the torus and give examples of groups acting on the torus by diffeomorphisms isotopic to the identity that cannot be extended to an action on a bounding 3-manifold. This solves a question raised in the previous work of the authors.

Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions

We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and every free product Γof n infinite amenable groups, the factor K^Γ/K of the Bernoulli action of Γon K^Γ by the diagonal action of K, is isomorphic with a Bernoulli action of Γ.

Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.