Quasi-isometric rigidity of higher rank S-arithmetic lattices

Quasi-isometric rigidity of piecewise geometric manifolds

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus …

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.

Veech群的扩展 II：分层双曲性与拟等距刚性

Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity

本研究针对闭曲面映射类群中的格点Veech群，证明其π₁S-扩展群为分层双曲群，进而得出这类扩展群具有拟等距刚性的结论，为Veech群扩展的几何性质研究提供了新结果。

Quasi-isometric rigidity of a class of right-angled Coxeter groups

We establish quasi-isometric rigidity for a class of right-angled Coxeter groups. Let $Γ_1,Γ_2$ be joins of finite generalized thick $m$-gons with $m\geq 3$. We show that the corresponding right-angled Coxeter groups are quasi-isometric if and only if $Γ_1,Γ_2$ are isomorphic. We also give a construction of commensurable right-angled Coxeter groups.

Day's fixed point theorem, Group cohomology and Quasi-isometric rigidity

In this note we explain how Day's fixed point theorem can be used to conjugate certain groups of biLipschitz maps of a metric space into special subgroups like similarity groups. In particular, we use Day's theorem to establish Tukia-type theorems and to give new proofs of quasi-isometric rigidity results.

扩展容许群的拟等距刚性

Quasi-isometric rigidity of extended admissible groups

本文引入了包含非几何3-流形基本群和Croke-Kleiner容许群的扩展容许群类，并证明了该群类具有拟等距刚性。

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

曲面群图的拟等距刚性研究

On the quasi-isometric rigidity of graphs of surface groups

该研究针对具有仅含刚性闭曲面基本群顶点的循环JSJ分解的字双曲群$Γ$，证明了所有与$Γ$拟等距的群均与$Γ$抽象可公度，推进了几何群论中拟等距刚性问题的研究。

整数的准等距刚性：一个初等导论

Quasi-isometric rigidity of the integers: an elementary primer

本文提供了一个初等证明，展示任何有限生成且准等距于整数的群都是整数的虚群。

Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple Lie groups

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $Γ$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.

Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.

幂零李群的Tukia型定理与可解群的拟等距刚性

A Tukia-type theorem for nilpotent Lie groups and quasi-isometric rigidity of solvable groups

本文研究Carnot-by-Carnot群的一致拟共形群，证明在诱导作用余紧的条件下可共轭到共形群，且该结果可用于研究某类可解群的拟等距刚性。

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。

Quasi-isometric rigidity of subgroups and Filtered ends

Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P\leq G$; is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions for a positive answer. The article consider pairs of the form $(G,\mathcal{P})$ where $G$ is a finitely generated group and $\mathcal{P}$ a finite collection of subgroups, there is a notion of quasi-isometry of pairs, and quasi-isometrically characteristic collection of subgroups. A subgroup is qi-characteristic if it belongs to a qi-charact…

树乘积中随机子集的拟等距刚性

Quasi-isometric rigidity for random subsets in products of trees

本文证明了从两棵正则树的乘积空间中的随机子集到其自身的拟等距嵌入的刚性结果，推广了Eskin的高秩非均匀格点的拟等距刚性，并描述了随机样本的自拟等距嵌入，同时证明两个独立样本几乎必然非拟等距。

On Rigidity of $S$-Arithmetic Kac-Moody Groups

In this article we investigate rigidity properties of $S$-arithmetic Kac-Moody groups in characteristic $0$.

Embeddability and quasi-isometric classification of partially commutative groups

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups $\mathbb{G}(Δ)$ and $\mathbb{G}(Γ)$ are quasi-isometric, then $\mathbb{G}(Δ)$ is a (nice) subgroup of $\mathbb{G}(Γ)$ and vice-versa. We show that the conjecture holds for all known cases of quasi-isometric classification of partially commutative groups, namely for the classes of $n$-tress and atomic graphs. As in the classical Mostow rigidity theory for irre…

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…

Almost isometric actions, property $(T)$, and local rigidity

Let $Γ$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $Γ$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in \cite{FM2}. One definition of property $(T)$ is that a group $Γ$ has property $(T)$ if every isometric $Γ$ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with proper…

Rigidity of warped cones and coarse geometry of expanders

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.

Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.

Large-scale conformal rigidity in dimension three

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the hyperbolic plane with the real line are large-scale conformally rigid. This implies new characterizations of groups that can act properly, cocompactly by isometries on those manifolds.

Quasi-isometries and rigidity of solvable groups

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R …

Profinite non-rigidity of arithmetic groups

We show that for a typical high rank arithmetic lattice $Γ$, there exist finite index subgroups $Γ_{1}$ and $Γ_{2}$ such that $Γ_{1} \not\simeq Γ_{2}$ while $\widehat{Γ_{1}} \simeq \widehat{Γ_{2}}$. But there are exceptions to that rule.

算术格的非有限刚性及Kähler性质

Profinite Non-Rigidity of Arithmetic Lattices and the Kähler Property

本文研究算术格的非有限刚性，刻画了允许非同构实形式的高秩李群绝对Dynkin类型，并证明有限呈现剩余有限群的Kähler性质不由其有限完备化决定。

Quasi-isometric classification of non-geometric 3-manifold groups

We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.

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…

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semisimple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

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].

First order rigidity of non-uniform higher rank arithmetic lattices

If $Γ$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example, $SL_n(\mathbb{Z})$, $n \geq 3$) and $Λ$ is a finitely generated group that is elementarily equivalent to $Γ$, then $Λ$ is isomorphic to $Γ$.

映射类群模扭转幂的刚性

Rigidity of mapping class groups mod powers of twists

本文研究带孔球面映射类群模Dehn扭转幂的商，证明其自同构与拟等距刚性，并发展分层双曲空间技术以给出裤子图刚性的新证明。

多项式增长灯泡群的拟等距刚性

Quasi-isometric rigidity for lamplighters with lamps of polynomial growth

本文证明多项式增长灯泡群的拟等距具有测度缩放刚性，缩放因子由增长次数决定，并应用于构造非双Lipschitz等价的群对及迭代圈积分类，首次给出灯泡刚性群例子。

Rigidity of commuting affine actions on reflexive Banach spaces

We give a simple argument to show that if α is an affine isometric action of a product G x H of topological groups on a reflexive Banach space X with linear part π, then either π(H) fixes a unit vector or α|G almost fixes a point on X. It follows that any affine isometric action of an abelian group on a reflexive Banach space X, whose linear part fixes no unit vectors, almost fixes points on X.

Modular Embeddings and Rigidity for Fuchsian Groups

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $Γ_1$, $Γ_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colonΓ_1\toΓ_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automorphism of $\mathrm{PGL}(2,\mathbb{R})$.

Quasi-isometries of rank one S-arithmetic lattices

We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator.

映射类群的随机商在拟等距下是刚性的

Random quotients of mapping class groups are quasi-isometrically rigid

本文证明了曲面映射类群随机商的拟等距刚性，推广了Ivanov定理，并利用层级双曲群技术证明弱等距性及自同构群与群本身重合。

Commability of groups quasi-isometric to trees

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consis…

大类型Artin群的刚性及分类结果

Rigidity and classification results for large-type Artin groups

本文计算了许多大类型Artin群交集图的自同构群，并以此为应用推导出包括外自同构群、可公度性分类及拟等距、测度等价、轨道等价、格嵌入和交叉积von Neumann代数刚性等一系列分类与刚性结果。

拉马努金复形的准等距嵌入

Quasi-isometric embeddings of Ramanujan complexes

本文研究了不同素数构造的拉马努金复形在大尺度几何上的区别，证明不同素数对应的复形无法相互准等距嵌入，利用了Box空间刚性和欧几里得建筑刚性等工具。

Quasi-actions on trees: research announcement

We develop a battery of tools for studying quasi-isometric rigidity and classification problems for splittings of groups. The techniques work best for finite graphs of groups where all edge and vertex groups are coarse PD groups. For example, if Gamma is a graph of coarse PD(n) groups for a fixed n, if the Bass-Serre tree of Gamma has infinitely many ends, and if H is a finitely generated group quasi-isometric to pi_1(Gamma), then we prove that H is the fundamental group of a graph of coarse PD(n) groups, with vertex and edge groups quasi-isometric to those of Gamma. We also have quasi-isometr…

Quasi-Isometric Embeddings of Symmetric Spaces

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,\mathbb R)$ into $Sp(2(n-1),\mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that ma…

Local Rigidity Of Uniform Lattices

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in $\text{Isom}(X)$ where $X$ is a proper $\text{CAT}(0)$ space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang's finiteness theorem for certain non-positively curved metric spaces.

On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.

使所有可数群成为准等距群

Realising all countable groups as quasi-isometry groups

本文通过构造新的强准等距刚性空间，证明了所有可数群都可以作为准等距群出现，且当群为超几何群时，所构造的空间也是超几何度量空间。

算术群与profinite群的范数刚性

Norm rigidity for arithmetic and profinite groups

本文证明具有有界初等生成的 ${\rm{SL}}_n(A)$ 的共轭不变范数具离散或预紧二分性，并将其与算术群及profinite群的刚性结果相关联，推导出对非剩余有限中心扩张近似的约束。

Poincaré inequalities and rigidity for actions on Banach spaces

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan's property (T) and is equivalent to the fact that $H^1(G,π)=0$ for every isometric representation $π$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we provide examples of groups, which satisfy such conditions and for which $H^1(G,π)$ vani…

Local-to-Global-rigidity of lattices in $SL_n(\mathbb{K})$

A vertex-transitive graph $\mathcal{G}$ is called Local-to-Global rigid if there exists $R>0$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $\mathcal{G}$ is covered by $\mathcal{G}$. An example of such a graph is given by the Bruhat-Tits building of $PSL_n(\mathbb{K})$ with $n\geq 4$ and $\mathbb{K}$ a non-Archimedean local field of characteristic zero.. In this paper we extend this rigidity property to a class of graphs quasi-isometric to the building including torsion-free lattices of $SL_n(\mathbb{K})$. The demonstration is the occasion to…

Rigidity results for certain 3-dimensional singular spaces and their fundamental groups

In this paper, we introduce a particularly nice family of locally CAT(-1) spaces, which we call hyperbolic P-manifolds. For $X^3$ a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of $X^3$ are characterized topologically. Straightforward consequences include a version of Mostow rigidity, as well as quasi-isometric rigidity for these spaces.

子图排除与平面刚性

Minor exclusion and planar rigidity

本文证明连通局部有限拟传递图若拟等距于排除可数团子图的图则拟等距于平面图，进而得到有限生成群为虚平面的等价条件，完善了正则映射相关的群论结论。

Warped cones, (non-)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…