折叠中位图

Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…

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.

为何应以中值图取代CAT(0)立方复形

Why CAT(0) cube complexes should be replaced with median graphs

该研究针对几何群论中的术语使用问题，提出应以中值图替代CAT(0)立方复形的表述，并阐述了相关依据与动因。

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…

Translation lengths in crossing and contact graphs of (quasi-)median graphs

Given a quasi-median graph $X$, the crossing graph $ΔX$ and the contact graph $ΓX$ are natural hyperbolic models of $X$. In this article, we show that the asymptotic translation length in $ΔX$ or $ΓX$ of an isometry of $X$ is always rational. Moreover, if $X$ is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph $X$ is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contac…

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。

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。

等价关系的树状图、墙结构和中位数图

Tree-like graphings, wallings, and median graphings of equivalence relations

本文证明了大尺度几何具有“树状”特征的局部有限 Borel 图会诱导可树化等价关系，并基于 proper wallings 与中位数图的 Stone 对偶性，将该结论进一步扩展至 Borel 度量空间并强化为超有限性。

Finite automata for Schreier graphs of virtually free groups

The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.

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。

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。