Hyperbolic groups admit proper affine isometric actions on $l^p$-spaces

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…

Asymptotically isometric metrics on relatively hyperbolic groups and marked length spectrum

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO(n,1). This partly verifies a conjecture by Margulis. In the case of hyperbolic groups/spaces, our result generalizes a theorem by Furman and a theorem by Krat. We discuss an application to the isospectral problem for the length spectrum of Riemannian manifolds. The positive answer to this problem has been known for several cases. All of them have hyperbolic fundamental groups. We do not solve the isospectral problem in the original se…

Proper isometric actions of hyperbolic groups on $L^p$-spaces

We show that every non-elementary hyperbolic group $\G$ admits a proper affine isometric action on $L^p(\bd\G\times \bd\G)$, where $\bd\G$ denotes the boundary of $\G$ and $p$ is large enough. Our construction involves a $\G$-invariant measure on $\bd\G\times \bd\G$ analogous to the Bowen - Margulis measure from the CAT$(-1)$ setting, as well as a geometric cocycle à la Busemann. We also deduce that $\G$ admits a proper affine isometric action on the first $\ell^p$-cohomology group $H^1_{(p)}(\G)$ for large enough $p$.

不与曲线复形拟等距的双曲空间

Hyperbolic spaces not quasi-isometric to curve complexes

本文提出了一个判定条件，用于排除双曲空间与非例外曲面曲线复形拟等距的可能性，并证明了该结果适用于弧复形、圆盘复形等多种双曲复形。

Relatively hyperbolic groups: geometry and quasi-isometric invariance

In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic triangle of a central left coset of peripheral subgroup.

Spaces with labelled partitions and isometric affine actions on Banach spaces

We define the structure of spaces with labelled partitions which generalizes the structure of spaces with measured walls and study the link between actions by automorphisms on spaces with labelled partitions and isometric affine actions on Banach spaces, and more particularly, on $L^p$ spaces. We build natural spaces with labelled partitions for the action of various contructions of groups, namely: direct sum; semi-direct product; wreath product and amalgamated free product. We apply this to prove that the wreath product of a group with property $PL^p$ by a group with Haagerup property has pro…

Fixed point-free isometric actions of topological groups on Banach spaces

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope of the universal Urysohn metric space, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isome…

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。

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…

Warped cones and proper affine isometric actions of discrete groups on Banach spaces

Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped cones $\mathcal{O}_Γ(M)$ admit a fibred coarse embedding into a $L_p$-space ($1\leq p<\infty$) if and only if the discrete group $Γ$ admits a proper affine isometric action on a $L_p$-space. This actually holds for any class of Banach spaces stable under taking Lebesgue-Bochner $L_p$-spaces and ultraproducts, e.g., uniformly convex Banach spaces or Banach spaces with nontrivial type. It follows that…

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples includ…

广义Baumslag-Solitar群与分层双曲群

Generalised Baumslag-Solitar groups and Hierarchically Hyperbolic Groups

本文研究任意维数广义Baumslag-Solitar群在双曲空间上的等距作用，证明实质上成为分层双曲群及性质（QT）在拟等距下均非不变量。

Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the…

Discretisable quasi-actions I: Topological completions and hyperbolicity

We define and develop the notion of a discretisable quasi-action. It is shown that a cobounded quasi-action on a proper non-elementary hyperbolic space $X$ not fixing a point of $\partial X$ is quasi-conjugate to an isometric action on either a rank one symmetric space or a locally finite graph. Topological completions of quasi-actions are also introduced. Discretisable quasi-actions are used to give several instances where commensurated subgroups are preserved by quasi-isometries. For example, the class of $\mathbb{Z}$-by-hyperbolic groups is shown to be quasi-isometrically rigid. We characte…

Quasi-isometrically rigid subgroups in right-angled Coxeter groups

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that, if two right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$ are quasi-isometric, then for any minsquare subgraph $Λ_1 \leq Γ_1$ there exists a minsquare subgraph $Λ_2 \leq Γ_2$ such that the right-angled Coxeter groups $C(Λ_1)$ and $C(Λ_2)$ are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a stu…

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。

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.

具有几何和几何有限准作用的双曲空间是对称的

Hyperbolic spaces with geometric and geometrically finite quasi-actions are symmetric

研究证明，若一个度量空间与有限生成群及具有上空的有限生成群空间准等距，则该空间与一维对称空间或实直线准等距。

Groups of piecewise isometric permutations of lattice points

Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in M, we consider the orbit Gp in M and define a notion of "G-polyhedral pieces" S in Gp. The objects of our interest are the groups pi(S) of all piecewise G-isometric permutations on S. In this paper we merely present the two most basic examples, and these play rather different roles: The case when G = PSL(2,Z) acting on the hyperbolic plane reveals that the "…

Quasi-hyperbolic planes in hyperbolic groups

The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.

acylindrically hyperbolic 群在 $\ell^1$ 上的作用

Actions of acylindrically hyperbolic groups on $\ell^1$

本文为具有双曲特征的群构造了 $\ell^1$ 和 $L^1$ 上的仿射一致 Lipschitz 作用，并证明对 acylindrically hyperbolic 群轨道无界，对剩余有限双曲群和映射类群轨道为真且诱导度量与字度量拟等距。

Groups of piecewise isometric permutations of lattice points or finitary rearrangements of tessellations

Through the glasses of didactic reduction: We consider a (periodic) tessellation $Δ$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $Δ$ we mean the process of cutting $M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation $Δ$. Such a rearrangement defines a permutation of the (centers of the) tiles of $Δ$, and we are interested in the group $PI(Δ)$ of all piecewise isometric rearrangements of $Δ$. In this paper we offer: a) An illustration of piecewise isometri…

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.

仿射仙人掌群的CAT(0)性质与双曲性

CAT(0) property and hyperbolicity of affine cactus groups

本文证明了所有度数的仿射仙人掌群都是CAT(0)群，并进一步证明了三次仿射仙人掌群AJ_3是双曲群。

Hyperbolic and cubical rigidities of Thompson's group V

In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson's group $V$ is hyperbolically elementary, and we deduce that it satisfies Property $(FW_{\infty})$, ie., every isometric action of $V$ on a finite-dimensional CAT(0) cu…

Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency $2^{\aleph_0}$ at every point. It implies that all these asymptotic cones are isometric depending neither on a manifold nor on an ultrafilter. It is also proved that the same R-tree can be isometrically embedded at infinity into such a manifold or into a non-abelian free group.

Equivariant embeddings of trees into hyperbolic spaces

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non…

Hyperbolic buildings, affine buildings and automatic groups

We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automatic. We turn our attention to affine buildings and consider a group $Γ$ which acts simply transitively and in a ``type-rotating'' way on the vertices of a locally finite thick building of type $\tilde A_n$. We show that $Γ$ is biautomatic, using a presentation of $Γ$ and unique normal form for each element of $Γ$, as described in ``Groups acting simply transitively on the vertices of a building of type $\tilde A_n$'' …

双曲群的特殊仿射表示

Special affine representations for hyperbolic groups

本文将特殊表示推广到容许互补序列的 Gromov 双曲群，证明自然约化上同调类的存在性，建立 Kuhn-Vershik 公式的类似物，并利用余循环等分布定理证明相应仿射作用不可约。

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.

A Limit Set Intersection Theorem for Graph of Relatively Hyperbolic Groups

Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and edge groups satisfy the limit set intersection property for conical limit points. This result is motivated by the work of Sardar for graph of hyperbolic groups.

Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings

We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M.~Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu's vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.

Locally compact groups with every isometric action bounded or proper

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, inc…

A product of trees as universal space for hyperbolic groups

We show that every Gromov hyperbolic group $\Ga$ admits a quasi-isometric embedding into the product of $(n+1)$ binary trees, where $n=\dim\di\Ga$ is the topological dimension of the boundary at infinity of $\Ga$.

双曲性与 $L^1$ 子空间上的一致 Lipschitz 仿射作用

Hyperbolicity and uniformly Lipschitz affine actions on subspaces of $L^1$

本文利用 Mineyev 的 $\mathbb{Q}$-双组合和 Balasubramanya 的拟树作用刻画，证明了双曲群和无端双曲群在 $L^1$ 空间子空间上分别具有真一致 Lipschitz 仿射作用和无界轨道的一致 Lipschitz 仿射作用。

Gromov 双曲空间的标记长度谱

Marked length spectra of Gromov hyperbolic space

本文证明了在群$Γ$的可比等距作用下，若两个粗测地完备的Gromov双曲空间具有相同的渐近标记长度谱且极限集相同，则它们是$Γ$-等变粗等距的。

Tangent bundles of hyperbolic spaces and proper affine actions on $L^p$ spaces

We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group $G$ acts on a metric measured space $X$ with a negatively curved tangent bundle, then $G$ acts on some $L^p$ space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when $X$ is a hyperbolic space.

Embedding mapping class groups into finite products of trees

We prove the equivalence between a relative bottleneck property and being quasi-isometric to a tree-graded space. As a consequence, we deduce that the quasi-trees of spaces defined axiomatically by Bestvina-Bromberg-Fujiwara are quasi-isometric to tree-graded spaces. Using this we prove that mapping class groups quasi-isometrically embed into a finite product of simplicial trees. In particular, these groups have finite Assouad-Nagata dimension, direct embeddings exhibiting $\ell^p$ compression exponent $1$ for all $p\geq 1$ and they quasi-isometrically embed into $\ell^1(\mathbb N)$. We deduce…

Bounding surface actions on hyperbolic spaces

We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly positive.

Quasi-hyperbolic planes in relatively hyperbolic groups

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3-manifolds. The key idea is to study quantitative geometric prop…

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。

Global and local boundedness of Polish groups

We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on Roelcke precompactness (precompactness of the lower uniformity) and Property (OB) (boundedness of all isometric actions on separable metric spaces). In particular, these properties are characterised by the orbit structure of isometric actions on metric spaces and isometric or continuous affine representations on separable Banach spaces.

Proper actions on $\ell^p$ spaces for relatively hyperbolic groups

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space. We prove that any quasiflat of dimension…

Quasi-isometries of relatively hyperbolic groups with an elementary hierarchy

Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be LERF and virtually toral relatively hyperbolic.

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.

Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji--Druţu--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0<p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occure when the linear part comes from a measure preserving action, or more generally a sta…

Simultaneous construction of hyperbolic isometries

Given isometric actions by a group G on finitely many δ-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a conjecture of Handel and Mosher regarding relatively fully irreducible subgroups and elements in the outer automorphism group of a free group.

Cusped spaces and quasi-isometries of relatively hyperbolic groups

A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in…

The co-surface graph and the geometry of hyperbolic free group extensions

We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\mathrm{Out}(\mathbb{F})$ quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of…

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.

用于莫比乌斯变换和群的双曲表示的核方法

Kernel method for Möbius transformations and hyperbolic representations of groups

研究群在双曲空间上等距作用，引入新核方法，可对\(\mathrm{PSL}_2(\R)\)相关表示分类，还能给出任意群此类表示共轭的充要条件。

Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary

Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

Word hyperbolic extensions of surface groups

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.