Hyperbolic isometries versus symmetries of links

Algebraic Characterization of Isometries of the Complex and Quaternionic Hyperbolic Planes

It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quaternionic hyperbolic space. Our result restricts to the complex case and provides another characterization of the isometries of the complex hyperbolic plane which is different from the characterization due to Giraud-Goldman. Two elements in a group G are said to be in the same z-cl…

Half-turn linked pairs of isometries of hyperbolic 4-space

In this paper we develop a complete theory of factorization for isometries of hyperbolic 4-space. Of special interest is the case where a pair of isometries is linked, that is, when a pair of isometries can be expressed each as compositions of two involutions, one of which is common to both isometries. Here we develop a new theory of hyperbolic pencils and twisting planes involving a new geometric construction, their half-turn banks. This enables us to give complete results about each of the pair-types of isometries and their simultaneous factorization by half-turns. That is, we provide geomet…

Integrable tautness of isometries of complex hyperbolic spaces

Consider $n \geq 2$. In this paper we prove that the group $\text{PU}(n,1)$ is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie groups of non-compact type. Additionally the tautness property implies a classification of finitely generated groups which are $\text{L}^1$-measure equivalent to lattices of $\text{PU}(n,1)$. More precisely, we show that $\text{L}^1$-measure equivalent groups must be extensions of lattices of $\text{PU}(n,1)$ by a finite group.

关于双曲3空间等距群的quandle

On the quandles of isometries of the hyperbolic 3-space

本文通过构造与Kleinian群及其非平凡元素相关联的quandle $Q(Γ, γ)$，将纽结quandle推广到双曲纽结，并建立了到$\mathrm{PSL}(2, \mathbb{C})$共轭quandle的具有离散像的单射同态。

Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups

The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated …

高亏格曲面精细曲线图的双曲等距

Hyperbolic isometries of the fine curve graph of higher genus surfaces

该研究针对闭双曲曲面上同痕于恒等映射的同胚，证明了其在精细曲线图上的双曲作用、相对于有限不变集同痕于pseudo-Anosov映射、遍历同调旋转集非空内部这三者的等价性。

Reversible Quaternionic Hyperbolic Isometries

Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the $n$-dimensional quaternionic hyperbolic space. Let $\mathrm{PSp}(n,1)$ be the isometry group of $\textbf{H}_{\mathbb H}^n$. In this paper, we classify reversible and strongly reversible elements in $\mathrm{Sp}(n)$ and $\mathrm{Sp}(n,1)$. Also, we prove that all the elements of $\mathrm{PSp}(n,1)$ are strongly reversible.

$z$-Classes of Isometries of The Hyperbolic Space

Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of "d…

On discreteness of subgroups of quaternionic hyperbolic isometries

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_{\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{\bf H}_{\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \in G$ the two generator subgroup $\langle f, g f g^{-1} \rangle$ is d…

Two Generator groups acting on the complex hyperbolic plane

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.

自由 Klein 群的对称分解与双曲位移

Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

研究自由 Klein 群中由对称等距子集产生的双曲位移，证明二生成元群在长度至多 k 时的统一位移下界，并对纯双曲有限生成自由群提出进一步下界猜想。

将伪阿诺索夫映射类计数为弱收缩等距映射

Counting pseudo-Anosovs as weakly contracting isometries

本文证明在有限型双曲曲面映射类群及更广的层级双曲群的Cayley图中，伪阿诺索夫映射类或Morse元素是普遍的，并建立了拟等距不变量计数理论。

Decomposition of complex hyperbolic isometries by involutions

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $λ$, $|λ|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most four involutions and a complex $k$-reflection, $k \leq 2$. Along the way, we prove that every element in ${\rm SU}(n)$ is a product of four or five involutions according as $n \neq 2 \mod 4$ or $n = 2 \mod 4$. We also give an easy proof of the result that every holomorphic isometry …

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…

Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct…

A Complex Quaternion Model for Hyperbolic 3-Space

In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in which both points and isometries are expressed as complex quaternions, analogous to Hamilton's famous theorem on Euclidean rotations. We use this to give new computational tools for studying isometries of hyperbolic 2- and 3-space. We also give a generalization to other quaternion algebras.

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of…

Random walks on weakly hyperbolic groups

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with …

Cusp transitivity in hyperbolic 3-manifolds

In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove a conjecture of Vogeler that there is a largest $k$ for which such $k$-transitive actions exist, and that for each $k \geq 3$, there is an upper bound on the possible number of cusps.

Infinitesimal gluing equations and the adjoint hyperbolic Reidemeister torsion

We establish a link between the holomorphic derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. Moreover, we provide a geometric reformulation of the non-abelian Reidemeister torsion corresponding to the adjoint of the monodromy representation of the hyperbolic structure. These results are then applied to the study of the `1-loop Conjecture' of Dimofte-Garoufalidis, which we generalize to arbitrary 1-cusped hyperbolic 3-manifolds. We rigorously verify the generalized co…

Gromov双曲空间与Teichmüller空间上随机等距变换平移长度的线性增长

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

该研究针对Gromov双曲空间等距群上的随机游走，证明无矩条件下非初等随机游走平移长度至少线性增长，有限一阶矩下增长率等于漂移，还将技术推广到Teichmüller空间，在有限一阶矩下证明谱定理，拓展了已有结果。

Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type $[m_1,m_2,0]$

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements.

Complex Hyperbolic Triangle Groups of Type $[m,m,0;3,3,2]$

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with pairwise distances $m,m,0$. For fixed $m,$ the parameter space of such groups is of real dimension one. We determine intervals in this parameter space that correspond to discrete and to non-discrete triangle groups.

熵与体积 via 海格劳德图

Entropy versus volume via Heegaard diagrams

该研究通过海格劳德弗洛尔同调和双曲几何，建立了双曲3流形的熵与体积之间的不等式关系，改进了Kojima的结果。

Nielsen methods and groups acting on hyperbolic spaces

We show that for any positive integer $n$ there exists a constant $C(n)>0$ such that any $n$-generated group $G$, which acts by isometries on a $δ$-hyperbolic space (with $δ>0$), is either free or has a nontrivial element with translation length at most $δC(n)$.

Lie Ball as Tangent Space to Poincare Space

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In the case of hyperbolic space, we exhibit an equivariant diffeomorphism between $TM$ and the fourth symmetric complex domain of E. Cartan, also known as the Lie ball. The closure of the Lie ball appears as a horospheric compactification of the tangent bundle to hyperbolic space, an…

包含抛物等距的群的离散性准则

A discreteness criterion for groups containing parabolic isometries

本文提出一个几何方法证明双曲空间中含抛物等距的群的离散性准则，推广了Shimizu-Leutbecher和Jorgensen的结果，并改进了高维的渐近界。

The hyperbolic meaning of the Milnor-Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor--Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor's function as equalities, with the deficiency from eq…

二生成元无挠 Klein 群与双曲位移

Two-Generator Free Kleinian Groups and Hyperbolic Displacements

本文针对无挠离散 Klein 群改进 $\log 3$ 位移定理，证明在相同条件下可从 $ξ$、$η$、$ξη$ 中找到等距，使任一点位移至少为 $(1/2)\log(5+3\sqrt{2})$。

Counting loxodromics for hyperbolic actions

Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the Gromov boundary $\partia…

Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces

The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bound…

Pattern Rigidity and the Hilbert-Smith Conjecture

In this paper we initiate a study of the topological group $PPQI(G,H)$ of pattern-preserving quasi-isometries for $G$ a hyperbolic Poincare duality group and $H$ an infinite quasiconvex subgroup of infinite index in $G$. Suppose $\partial G$ admits a visual metric $d$ with $dim_H < dim_t +2$, where $dim_H$ is the Hausdorff dimension and $dim_t$ is the topological dimension of $(\partial G,d)$. a) If $Q_u$ is a group of pattern-preserving uniform quasi-isometries (or more generally any locally compact group of pattern-preserving quasi-isometries) containing $G$, then $G$ is of finite index in $…

纯loxodromic自由Kleinian群中长度为$1$的等距与迹不等式

Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

本文推广了PSL₂(C)子群的离散性判据，证明纯loxodromic自由Kleinian群中满足特定位移条件时存在迹不等式，常数由KKT理论确定。

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…

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.

Counting problems in graph products and relatively hyperbolic groups

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.

Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

Virtual Amalgamation of Relatively Quasiconvex Subgroups

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.

Cauchy-compact flat spacetimes with extreme BTZ

Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries $\mathrm{PSL}_2(\R)$, admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with group of isometries $\mathrm{PSL}_2(\R)\ltimes \RR^3$, admitting finitely many ends whose neighborhoods are foliated by cusps. We prove a Theorem akin to the classical parametrization result for finite volume …

Hierarchically hyperbolic groups are determined by their Morse boundaries

We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a 2-stable, quasi-möbius homeomorphism between their Morse boundaries. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental group…

Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n > 1. The group G is acting on the universal cover of M, which is isometric to H^n_C. It …

On the Limit Set of Root Systems of Coxeter Groups acting on Lorentzian spaces

The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection group of isometries on a hyperbolic space; the accumulation set of any of its orbits is then classically called the limit set of W. In this article we show that the limit roots of a Coxeter group W acting on a Lorentzian space is equal to the limit set of W seen as a discrete reflect…

Properly discontinuous actions versus uniform embeddings

Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $ρ: G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting properly on a contractible $n$-manifold and uniformly embedding into a contractible $n$-manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible $3$-manifold but only virtually act on a contractible…

Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $δ$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $δ$ and $H$ alone. From these results, we draw homo…

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…

Geometry of the Gromov product : Geometry at infinity of Teichmüller space

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller …

Contracting elements and random walks

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in $Out(F_n)$, $n\geq 3$. We prove that any simple random walk…

Geometric survey of subgroups of mapping class groups

The theme of this survey is that subgroups of the mapping class group of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of the Teichmuller space of S, just as discrete subgroups of the isometries of hyperbolic space can be studied via their action on compactified hyperbolic space.

A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-Manifolds

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multi…

关于复双曲三角形群的类型

On the type of a complex hyperbolic triangle group

本文研究由三个复对合生成的离散复双曲等距群，给出其规范形式，并改进Schwartz猜想，在参数空间中确定两个词哪个先变为椭圆型。

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 …

凸-紧致表示到无限维双曲空间的等距群

Convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space

本文研究了无限维双曲空间等距群中有限生成群的凸-紧致表示性质，并通过弯曲技术获得非共轭于特定分类的表面群表示。

柱面双曲群中的增长间隙与指数一般性

Growth gaps and exponential genericity in acylindrically hyperbolic groups

研究柱面双曲群，证明对其每个有限生成集，非WPD元素集指数增长率更小，即WPD元素指数一般，还证明了该群的增长紧性和余增长紧性。

The Poisson boundary of hyperbolic groups without moment conditions

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.