Quasi-isometry rigidity of groups
Cornelia Drutu
本文综述了有限生成群，特别是相对双曲群的拟等距分类的最新进展。
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
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Quasi-isometric rigidity for PSL(2,Z[1/p])
我们证明了PSL（2，Z[1/p]）给出了第一个彼此不拟等距但具有相同拟等距群的群的例子。也就是说，PSL（2，Z[1/p]）和PSL（2，Z[1/q]）不是拟等距群，除非p=q，并且与p无关，PSL（2，Z[1/p]）的拟等距群是PSL（2，Q）。此外，我们还利用拟等距型刻画了PSL（2，Z[1/p]）在所有有限生成群中的唯一性.
We prove that the every quasi-isometry of Teichmüller space equipped with the Teichmüller metric is a bounded distance from an isometry of Teichmüller space. That is, Teichmüller space is quasi-isometrically rigid.
Volume and rigidity of hyperbolic polyhedral $3$-manifolds
研究了双曲空间中理想和超理想四面体等距胶合的$3$ -流形上双曲锥度量的刚性。这些度量将被称为理想和超理想双曲多面体度量。证明了超理想双曲多面体度规在等距下由曲率决定，修饰理想双曲多面体度规在等距和修饰变化下由曲率决定.证明中使用的主要工具是体积函数的Fenchel对偶。
We show that every $\mathbb R$-linear surjective isometry between the cotangent spaces to the Teichmüller space equipped with the Thurston norm is induced by some isometry between the underlying hyperbolic surfaces, which is an analogue of Royden's theorem concerning the Teichmüller metric.
Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $δ$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend the rank-rigidity result of Caprace-Sageev to any lattice $ Γ\leq$ Isom(X).
Marked length spectrum rigidity in groups with contracting elements
本文研究了粗几何背景下的标记长度谱刚性问题，证明具有收缩性质的群作用在相同标记长度谱下轨道映射为粗等距，并推广了双曲群的相关结论。
Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the coco…
A finitely-presented solvable group with a small quasi-isometry group
我们展示了一族无限的、有限表现的、幂零的阿贝尔群。这个群族的每个成员都是一个与Baumslag-Solitar群相关的可解S-算术群，并且这些群中的每个群都有一个准等距群，它实际上是一个可解实李群和一个可解p-adic李群的乘积。此外，我们提出了一个多环群的准等距群是一个可解的真正的李群的候选人，我们介绍了一个准等距刚性可解群，这是不确定的候选人。我们还记录了一些关于点灯人群体的大尺度几何的描述。
Quasi-isometry classification of certain graph $2$-braid groups and its applications
本文利用交复形理论完成周长≤1的图2-辫群拟等距分类，给出其与右角Artin群拟等距的充分条件，并提出判定两棵树4-辫群拟等距的算法。
Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry
该研究通过准等距性证明右角Artin群的可积分度量等价刚性，并揭示其超刚性定理及局部紧致群的结构限制。
We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n>1, we give examples of right-angled Coxeter groups which are thick o…
We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditio…
Commensurability and quasi-isometry classification for one vertex one loop tubular groups
本文对单顶点单环管状群按边映射交数分类：非零交数时全体可公度且拟等距；零交数时有两个拟等距类和无穷可公度类，两类间不拟等距。
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian manifolds which are almost-isometric have the same volume growth entropy. We establish various rigidity results as applications.
Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. The Milnor-Schwarz lemma implies that groups with a common model geometry are quasi-isometric; however, the converse is false in general. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually …
For a proper geodesic metric space $X$, the Morse boundary $\partial_*X$ focuses on the hyperbolic-like directions in the space $X$. It is a quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. In this paper, we investigate additional structures on the Morse boundary $\partial_*X$ which determine $X$ up to a quasi-isometry. We prove that, for $X$ and $Y$ proper, cocompact spaces, a homeomorphism $f$ between their Morse boundaries is induced by a quasi-isometry if and only if $f$ and $f^{-1}$ are bihölder, or quasi-symmet…
Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag-Solitar groups
我们说一个群具有性质$R_{\infty}$，如果任何群自同构都有无穷多个扭共轭类。Fel'shtyn和Goncalves证明了可解Baumslag-Solitar群BS（1，m）具有性质$R_{\infty}$ .我们定义了这些群的可解推广$Γ(S)$，并证明了它具有性质$R_{\infty}$。证明了这些群的性质$R_{\infty}$是几何的，即任何与$Γ(S)$拟等距的群也具有性质$R_{\infty}$.
Rigidity on horocycles and hypercycles
本文证明双曲平面中保持极限圆或超圆的映射必为等距，并推广至图的抽象自同构由地震映射或等距诱导，揭示测地线与极限圆/超圆几何的差异。
Quasi-isometry invariants of weakly special square complexes
本文定义弱特殊方形复形万有覆盖的交复形作为拟等距不变量，用于研究二维右角Artin群和平面图2-辫群的拟等距分类，并证明两类群中各有无穷多个与右角Artin群拟等距或不等距。
Quasi-isometric rigidity of higher rank S-arithmetic lattices
证明了半单李群中不含秩1因子的S-算术格是拟等距刚性的。
The quasi-isometry invariance of the Coset Intersection Complex
本文研究了余弦交复形的准等距不变性，证明了其与群对的代数性质相关，并展示了其在特定群类中的几何特性。
Quasi-isometry classification of right-angled Artin groups II: several infinite out cases
针对哪类右角Artin群（RAAG）的拟等距分类与可公度分类一致的问题，本文确定两类外自同构群含相邻平延和部分共轭的无限RAAG，证明其拟等距等价于可公度，提出基于定义图的判断算法，并通过形变论证和立方化技术解决重构映射问题。
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
研究了非相对双曲群的几何。推广了Schwartz的一个结果，非相对双曲空间在相对双曲空间中的任何拟等距像都包含在单个外围子群的有界邻域中.这意味着一个群是相对双曲的，而其周边子群是非相对双曲的，则它是一个拟等距不变量。作为一个应用，阿廷群是相对双曲的当且仅当可自由分解。我们还引入了一个新的准等距不变量的度量空间称为度量厚，这是足够的度量空间是非双曲相对于任何非平凡的子集的集合。厚有限生成群包括：大多数曲面的映射类群;大多数自由群的外自同构群;某些Artin群;以及其他。高阶半单李群中的非均匀格是厚的，因此是非相对双曲的，与提供相对双曲群的激励性例子的秩1相反。映射类群是第一个例子nonrelatively双曲群有切割点在任何渐近锥面，解决了几个问题的Drutu和萨皮尔的结构相对双曲群。在群论之外，Teichmuller空间对于足够大的复杂性的曲面来说相对于Weil-Peterson度量是厚的，与Brock-Farb的双曲性结果相比，它的复杂性很低。
We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.
Quasi-isometric classification of graph manifold groups
证明了任意两个闭不可约非几何图流形的基本群是拟等距的。这回答了Kapovich和Leeb的一个问题。我们还利用某些有限的双色有色图对有边图流形的基本群的拟等距型进行了分类。一个推论是Artin群的拟等距分类，其表示图是树。特别是任何两个直角阿廷组的介绍图是树的直径至少为3是准等距，回答一个问题的Besthold;此外，这个准等距类不包括任何其他直角阿廷组。
Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II
设BS（1，n）= < a，b：aba^{-1}=b^n >.证明了任何与BS（1，n）拟等距的有限生成群（直到有限群）同构于BS（1，n）.我们还证明了实直线的拟相似性的任何一致群都是仿射群的bilipschitz共轭群。
Quasi-isometric rigidity for a product of lattices
研究证明非均匀一维晶格与 nil 晶格乘积的准等距刚性，并展示其扩展为 nilcentral 扩展
Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups
在这些课堂讲稿中，我们将Kevin Whyte最近的同调方法与Benson Farb和我自己开发的较旧的动力学方法结合起来，得到亏格至少为2的一次穿孔曲面S的映射类群MCG（S）的一个新的拟等距刚性定理：若K是与MCG（S）拟等距有限生成群，则存在具有有限核和有限指标象的同态K -> MCG（S）。这个定理是与Kevin Whyte联合提出的。
A structure theorem and left-orderability of a quotient of quasi-isometry group of the real line
本文研究实数直线拟等距群的商群，通过引入不变量分解其元素，证明了该商群是左序但非局部可指标的群。
Isometry groups of non-positively curved spaces: discrete subgroups
本文研究非正曲度量空间中的格。博雷尔密度是建立在该设置以及一种形式的莫斯托刚度。提供了平坦环面定理的逆定理。几何算术性的结果是在绕道抽象格的超刚性和算术性之后得到的。研究了格的剩余有限性。黎曼对称空间的特征在于CAT（0）空间之间的承认格在存在的抛物等距。
Geometry of the mapping class groups III: Quasi-isometric rigidity
设S是亏格为g的有向有限型曲面，有m个穿孔，其中3g-3 + m> 1.证明了S的映射类群M（S）是拟等距刚性的.本文还对Behrstock和Minsky的结果：M（S）的渐近锥的同调维数等于3g-3 + m给出了不同的证明。
Failure of quasi-isometric rigidity for infinite-ended groups
本研究证明了满足特定条件的无限端群不具备拟等距刚性，结合已有工作得出剩余有限多端双曲群拟等距刚性的充要条件，证明方法改编自Whyte的相关论证。
Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of fundamental groups of spaces in $\mathcal{X}_S$. In this paper, we characterize the abstract commensurability classes within $\mathcal{C}_S$ in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in $\mathcal{C}_S$ are quasi-isometric by exhibiting a bilipschitz …
Geometry and rigidity of mapping class groups
利用曲线复形的双曲性质，研究了映射类群MCG（S）的大尺度几何。我们证明了MCG（S）的任何自拟等距（除了少数零星的情形）都是离左乘有界距离的，从而得到了MCG（S）的拟等距刚性，即与MCG（S）拟等距的群几乎等于它（Hamenstadt用不同的方法证明了后一个定理）.作为我们的方法的一部分，我们得到了其他几个结构性的结果：树阶结构的描述的渐近锥面的MCG（S）;的曲线复投影映射从MCG（S）的本质子曲面的曲线复的产品的形象的特征;和建设的Sigma壳在MCG（S），一个类似的凸包。
Relative Rigidity, Quasiconvexity and C-Complexes
本文引入并研究了双曲度量空间对$(X,\JJ)$的相对刚性概念，其中1）$X$是双曲度量空间，$\JJ$是拟凸集的集合; 2）$X$是相对双曲群，$\JJ$是抛物集的集合; 3）$X$是一个高阶对称空间，而$\JJ$是一个极大平坦的等变集合。相对刚性可以粗略地描述为提升一个一致真映射，两个这样的$\JJ$到相应的$X$之间的准等距。一个相关的概念是$C$复合体，它是山雀复合体对这种环境的适应。我们如上所述证明了对$(X, \JJ)$的集合的相对刚性。这推广了Schwarz关于双曲空间中测地线对称型的一个结果。我们证明了一致真映射诱导了相应的$C$ -复形的同构.我们也给出了拟凸性的几个刻画。双曲群的子群。
Quasiconformal Maps between Bowditch Boundaries of Relatively Hyperbolic Groups
研究相对双曲群的拟等距分类，核心方法是通过鲍迪奇边界引入拟共形映射概念，主要贡献是证明相对双曲群间粗保尖点的拟等距与粗保horoball阴影的鲍迪奇边界拟共形同胚相互诱导。
Coarse embeddings of products of trees as quasi-isometry invariants
研究茂密树乘积能粗嵌入到多种群中的最大因子数，利用此区分相关群并排除粗嵌入，基于树乘积到分层双曲空间的粗嵌入一般陈述得出这些应用。
A survey on quasi-isometries of pairs: invariants and rigidity
本文调查了群与子群配对的准等距性质，探讨了相关不变量和刚性理论。
Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups
本文是[EFW 2]的继续，我们完成了Sol和点灯群的拟等距刚性的证明。结果在[EFW 1]中公布。
We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomology of bounded geometry contact manifolds.
A Combinatorial Criterion and Center for the quasi-isometry groups of Euclidean spaces
本文引入$PL_δ$-同胚概念，给出基于单纯形顶点和边的组合判据判断分段线性同胚是否为拟等距，并证明$\mathbb{R}^n$的拟等距群$QI(\mathbb{R}^n)$的中心平凡。
Quasisymmetric universality, quasi-isometric classification and topological rigidity of Kleinian groups
该研究针对克莱因群极限集，通过刻画几何有限极限集的拟对称普适性，实现有限生成克莱因群的拟等距分类及几乎拓扑刚性几何有限克莱因群的分类
Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity
本文引入Patterson-Sullivan系统，研究其在随机游走和熵刚性中的应用，证明了Tukia的测度边界刚性定理的推广，并探讨了Patterson-Sullivan测度的奇异性猜想。
Isometry groups of infinite-genus hyperbolic surfaces
本文对无平面端的无限亏格二维流形的常曲率黎曼度量等距群给出近乎完全分类，证明具自相似端空间的流形可实现所有可数群为等距群，并推导相关群的性质阻碍与映射类群代数刚性结果。
In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometric rigidity for fundamental groups of cusp-decomposable manifolds.
Quasisymmetric rigidity of Sierpinski carpets $F_{n,p}$
研究了一类新的正方形Sierpienski地毯$F_{n,p}$（$5\leq n, 1\leq p<\frac{n}{2}-1$）在$\mathbb{S}^2$上的性质，它与标准Sierpienski地毯不是拟对称等价的.证明了每个$F_{n,p}$的拟对称自映射群是欧氏等距群。我们还建立了$F_{n,p}$和$F_{n',p'}$是拟对称等价的当且仅当$(n,p)=(n',p')$。
Quasi-isometric rigidity of three manifold groups
本文证明了有限生成Kleinian群类和三维流形群类具有拟等距刚性。
We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly determined by the translation lengths of images of elements of $π_1(M)$ represented by simple curves on the boundary of $M$. As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized intersection number is generated by the (…
Quasiconformal Rigidity of Negatively Curved Three Manifolds
本文研究了具有截面曲率$-b^2\le K\le -1$的无限体积三维流形的刚性，并讨论了它的生成基本群。特别地，我们将具有空耗散集的非线性生成基本群的Sullivan拟共形刚性推广到负变曲率三维流形。我们还推广了Hamenstädt或最近Besson-Courtois-Gallot的刚性，3-流形具有无限体积和几何无限的基本群。
Isometries, rigidity, and universal covers
我们对所有闭的非球面黎曼流形M进行了分类，M的万有覆盖具有不离散等距群。一个例子应用是定理，任何这样的M字双曲基本群必须等距到一个负弯曲，局部对称流形。另一个应用是所有可收缩的黎曼流形的分类覆盖紧和（非紧的，完整的）有限体积流形。也有应用到霍普夫猜想，一个新的证明Kazhdan猜想（弗兰克尔定理）在复杂的几何等思想的证明来自李理论，同调理论的变换群，调和映射，和大规模的几何。非非球面的情况下，也给出了一个扩展。
A cusp-decomposable manifold is a manifold constructed from a finite number of complete, negatively curved, finite volume manifolds and identifying the boundaries of truncated cusps by diffeomorphisms. Using properties of the electric space of the universal cover of cusp-decomposable manifolds, we show that the inclusion of walls and pieces induces quasi-isometric embeddings. We also show that isomorphisms between fundamental groups of higher graph manifolds preserve the decomposition into pieces.
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