Hyperbolically embedded subgroups and quasi-isometries of pairs
Sam Hughes, Eduardo Martínez-Pedroza
该研究给出配对拟等距保持子群双曲嵌入性的技术条件，并将其应用于有限生成群柱形双曲性的拟等距与可公度不变性研究。
我们给出配对拟等距保持子群双曲嵌入性的技术条件，还考虑了其在有限生成群的柱形双曲性的拟等距与可公度不变性中的应用。
We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically embedded. We consider applications to the quasi-isometry and commensurability invariance of acylindrical hyperbolicity of finitely generated groups.
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Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs.
Let G be a group, H a hyperbolically embedded subgroup of G, V a normed G-module, U an H-invariant submodule of V. We propose a general construction which allows to extend 1-quasi-cocycles on H with values in U to 1-quasi-cocycles on G with values in V. As an application, we show that every group G with a non-degenerate hyperbolically embedded subgroup has dim H^2_b (G, l^p(G))=\infty for p\in [1, \infty). This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undisto…
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
We show that any infinite order element $g$ of a virtually cyclic hyperbolically embedded subgroup of a group $G$ is Morse, that is to say any quasi-geodesic connecting points in the cyclic group $C$ generated by $g$ stays close to $C$. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyperbolically embedded subgroups are quasi-convex. Finally, we give a definition of what it means for a collection of subspaces of a metric space to be hyperbolically embedded and we show that axes of pseudo-Anosovs are hyperbolically embedded in Teichmüller space endowed with the We…
Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_λ,λ\in Λ\} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_λ,λ\in Λ\} \cup \{Q\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_λ$, $λ\in Λ$, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementar…
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic gro…
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$…
A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph is a (non-elementary) quasi-tree and the action of G on the Cayley graph is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As a by-product, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.
Relative Dehn fuctions, hyperbolically embedded subgroups and combination theorems
本文针对群在有限图上的分裂，证明了双曲嵌入子群和相对Dehn函数良定义性在组合下的保持性，并给出相对Dehn函数的上下界，推广了已有结果。
A notion of quasiconvex subgroups in acylindrically hyperbolic groups
本文在含双曲嵌入子群的有限生成群中提出拟凸性概念，证明其在锥化尖点空间有一致拟凸常数、长Dehn填充下保持不变，并推广了Groves-Manning关于CAT(0)立方复形群作用的定理。
Whyte showed that any quasi-isometry between non-amenable groups is a bounded distance from a bijection. In contrast this paper shows that for amenable groups, inclusion of a proper subgroup of finite index is never a bounded distance from a bijection.
We prove that commensurizers of two-ended subgroups with at least three coends in one-ended, finitely presented groups are invariant under quasi-isometries. We discuss a variety of applications of this result.
Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any G-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we …
Conjugator length of locally compact groups of Euclidean isometries
研究欧几里得\(n\)维空间全等距群中特定局部紧子群\(H\)，通过分析其球面部分拓扑结构，证明其共轭长度函数有零增长、线性增长或无界三种情况，还指出仿射考克斯特群等共轭长度函数线性增长。
Isometries are ubiquitous in nature; isometries of discrete (quantized) objects---abstracted as the group of isometries of $\mathbb{Z}^n$ denoted by $\mathsf{ISO}(\mathbb{Z}^n)$---are important concepts in the computational world. In this paper, we compute various isometric invariances which mathematically are orbit-computation problems under various isometry-subgroup actions $H \curvearrowright \mathbb{Z}^n, H \leq \mathsf{ISO}(\mathbb{Z}^n)$. One computational challenge here is about the \emph{infinite}: in general, we can have an infinite subgroup acting on $\mathbb{Z}^n$, resulting in poss…
We generalize [Vav] to give sufficient conditions, primarily on coarse geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff distance from a subgroup. Using this result, we prove a partial converse to the Flat Torus Theorem for CAT(0) groups. Also using this result, we give sufficient conditions for subgroups and splittings to be invariant under quasi-isometries.
Let G be any finitely generated infinite group. Denote by K(G) the FC-centre of G, i.e., the subgroup of all elements of G whose centralizers are of finite index in G. Let QI(G) denote the group of quasi-isometries of G with respect to word metric. We observe that the natural homomorphism from the group of automorphisms of G to QI(G) is a monomorphism only if K(G) equals the centre Z(G) of G. The converse holds if K(G)=Z(G) is torsion free. We apply this criterion to many interesting classes of groups.
Pattern preserving quasi-isometries in lamplighter groups and other related groups
该研究针对三类半直积结构群族，探究粗置换Z子群陪集的拟等距在B上的诱导映射，发现点灯群情形下不满足仿射映射结论，但仍有部分共性，拓展了群的几何代数关联研究。
We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on them, and 3. prove a Schur-Zassenhaus type splitting result.
Let S be a closed, oriented surface of genus at least 2, and consider the extension 1 -> pi_1 S -> MCG(S,p) -> MCG(S) -> 1, where MCG(S) is the mapping class group of S, and MCG(S,p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S,p) which coarsely respects the cosets of the normal subgroup pi_1 S is a bounded distance from the left action of some element of MCG(S,p). Combined with recent work of Kevin Whyte this implies that if K is a finitely generated group quasi-isometric to MCG(S,p) then there is a homomorphism K -> MCG(S,p) with finite kernel and…
We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their JSJ decompositions. The first condition is that their JSJ decompositions have two-ended cylinder stabilizers. The second is that every factor in their JSJ decompositions is either `relatively rigid' or `…
A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$κ$-to-one in a natural sense for some $κ>0$. For non-amenable graphs, all quasi-isometries are quasi-$κ$-to-one for any $κ>0$, while for amenable ones there exists at most one possible such $κ$. For an amenable graph $X$, we show that the set of possible $κ$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a giv…
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。
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.
Quasi-Isometries for certain Right-Angled Coxeter Groups
本文针对有限展示、单端、二维直角考克斯特群，基于定义图构造柱面JSJ树，证明其二端边稳定子与定义图不含特定细分K4等价，并利用结构不变量给出由定义图决定的拟等距不变量，在无刚性顶点情形下进一步细化为完全拟等距不变量。
Lipschitz metric isometries between Outer Spaces of virtually free groups
本文将自由群有限指数覆盖诱导的 Lipschitz 等距嵌入及折叠路径保持性推广到虚拟自由群，并证明距离候选存在性、建立其 Outer Space 脊与 Krstić–Vogtmann 空间的形变收缩关系。
Ping-pong in Hadamard manifolds
本文针对负压缩 Hadamard 流形建立定量 Tits 二择一结果，证明两个非椭圆等距生成的非初等离散群含词长一致有界的秩 2 自由子群，并在双曲情形给出凸余紧性。
Scale Groups
本文证明了正则树等距群中固定端点且顶点传递的闭子群，与根树上的自复制群及具有正尺度的全不连通局部紧群元素之间存在对应关系。
An extension theorem for quasimorphisms
本文研究了双曲群的正规子群上准同态的扩展条件，并给出了稳定交换长度的双曲等价性结果。
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…
Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ …
We prove that if $H$ is a subgroup of index $n$ of any cyclic group $G$, then $G$ can be isometrically embedded in $(H^n, d_{_{Ham}}^n)$, thus generalizing previous results of Carlet (1998) for $G=\mathbb{Z}_{2^k}$ and Yildiz-Özger (2012) for $G=\mathbb{Z}_{p^k}$ with $p$ prime. Next, for any positive integer $q$ we define the $q$-adic metric $d_q$ in $\mathbb{Z}_{q^n}$ and prove that $(\mathbb{Z}_{q^n}, d_q)$ is isometric to $(\mathbb{Z}_q^n, d_{RT})$ for every $n$, where $d_{RT}$ is the Rosenbloom-Tsfasman metric. More generally, we then demonstrate that any pair of finite groups of the same…
A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.
Let T be a rooted tree and Iso(T) be the group of isometries of T. Using model-theoretic tools we study closed subgroups G of Iso(T) with respect to the number of conjugacy classes of Iso(T) having representatives in G.
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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…
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 $…
Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients in the regular representation is infinite dimensional. The result holds for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually split as a direct product.
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.
We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.
Discrete two-generator subgroups of ${\rm PSL_2}$ over non-archimedean local fields
本文研究非阿基米德局部域上 ${\rm PSL_2}$ 二元生成子群的离散性，给出充要条件与判定算法，并建立相关 ${\rm SL_2}$ 子群稠密性算法。
Quasi-isometry invariance of relative filling functions
本研究证明有限生成群与子群配对的相对Dehn函数具有拟等距不变性，揭示拟等距对几乎反正规性、锥化Cayley图精细性的保持作用，明确组合Dehn函数良定义的等价条件，给出双曲嵌入子群相关结论及Baumslag-Solitar群的特例结果。
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