Quasi-isometry rigidity of groups

Uncountably many quasi-isometry classes of groups of type $FP$

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 and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen

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

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large…

某些图2-辫群的拟等距分类及其应用

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。

通过准等距性研究右角Artin群的可积分度量等价刚性

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。

A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

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…

Rigidity Properties for Hyperbolic Generalizations

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…

管状群的可公度性与拟等距分类：单顶点单环情形

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。

Rigidity of Almost-Isometric Universal Covers

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.

Action rigidity for free products of hyperbolic manifold groups

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 …

Maps on the Morse boundary

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

We say that a group has property $R_{\infty}$ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property $R_{\infty}$. We define a solvable generalization $Γ(S)$ of these groups which we show to have property $R_{\infty}$. We then show that property $R_{\infty}$ is geometric for these groups, that is, any group quasi-isometric to $Γ(S)$ has property $R_{\infty}$ as well.

关于极限圆与超圆的刚性

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。

弱特殊方形复形的拟等距不变量

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

We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.

余弦交复形的准等距不变性

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。

右角Artin群的拟等距分类II：若干无限外自同构群情形

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

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…

On systoles and ortho spectrum rigidity

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

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of graph-manifolds with boundary in terms of certain finite two-colored graphs. A corollary is the quasi-isometry classification of Artin groups whose presentation graphs are trees. In particular any two right-angled Artin groups whose presentation graphs are trees of diameter at least 3 are quasi-isometric, answering a question of Bestvina; further, this quasi-isometry …

Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

非均匀一维晶格与 nil 晶格乘积的准等距刚性

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

In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once punctured surface S of genus at least 2: if K is a finitely generated group quasi-isometric to MCG(S) then there is a homomorphism K -> MCG(S) with finite kernel and finite index image. This theorem is joint with Kevin Whyte.

实数直线拟等距群的一个商群的结构定理与左序性

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

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

Geometry of the mapping class groups III: Quasi-isometric rigidity

Let S be an oriented surface of finite type of genus g with m punctures and where 3g-3+m>1. We show that the mapping class group M(S) of S is quasi-isometrically rigid. We also give a different proof of the following result of Behrstock and Minsky: The homological dimension of the asmyptotic cone of M(S) of S equals 3g-3+m.

无限端群的拟等距刚性失效

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。

Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

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

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S…

Relative Rigidity, Quasiconvexity and C-Complexes

We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove t…

相对双曲群的鲍迪奇边界之间的拟共形映射

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。

作为拟等距不变量的树的乘积的粗嵌入

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。

关于配对的准等距：不变量与刚性调查

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

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

Cup-products in L q,p -cohomology: discretization and quasi-isometry invariance

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.

欧几里得空间拟等距群的组合判据与中心

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。

克莱因群的拟对称普适性、拟等距分类与拓扑刚性

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

Patterson-Sullivan系统刚性及其在随机游走和熵刚性中的应用

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。

无限亏格双曲曲面的等距群

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。

Quasi-Isometric Rigidity of Cusp-Decomposable Manifolds

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 $F_{n,p}$

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that $F_{n,p}$ and $F_{n',p'}$ are quasisymmetrically equivalent if and only if $(n,p)=(n',p')$.

三维流形群的拟等距刚性

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。

Simple length rigidity for Kleinian surface groups and applications

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

In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature 3-manifolds. We also generalize the rigidity of Hamenstädt or more recently Besson-Courtois-Gallot, to 3-manifolds with infinite volume and geometrically infinite fundamental group.

Isometries, rigidity, and universal covers

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of all contractible Riemannian manifolds covering both compact and (noncompact, complete) finite volume manifold. There are also applications to the Hopf Conjecture, a new proof of Kazhdan's Conjecture (Frankel's Theorem) in complex geometry, etc. Ideas in the proof come from Lie the…

On the quasi-isometric rigidity of chambers and walls in cusp-decomposable manifolds

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.