Ilya Kapovich, Richard Weidmann
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 $δC(n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of Out$(F_n)$.
We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N distinct Nielsen classes of generating pairs. - Two pairs of elements of a fundamental group of an HNN-extension are Nielsen equivalent iff they are so for the obvious reasons.
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a free product of Guirardel--Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups ar…
We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction.
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 $CAT(0)$…
Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,..., b_k)$ and $w_i\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,a_k)$ and $(b_1,...,b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,..., u_k$ and $v_1,..., v_k$ the "once-stabilized" tuples $(a_1,..., a_k,1)$ and $(b_1,...,b_k,1)$ are not Nielsen equivalent in $G$. This provides a counter-example for a Wiego…
This is a (very subjective) survey paper for nonspecialists covering group actions on Gromov hyperbolic spaces. The first section is about hyperbolic groups themselves, while the rest of the paper focuses on mapping class groups and $Out(F_n)$, and the way to understand their large scale geometry using their actions on various hyperbolic spaces constructed using projection complexes. This understanding for $Out(F_n)$ significantly lags behind that of mapping class groups and the paper ends with a few open questions.
Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff's result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the id…
We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppos…
Stability for hyperbolic groups acting on boundary spheres
本文证明，当双曲群的 Gromov 边界为拓扑 n-球面时，其边界同胚作用具有动力学拓扑稳定性，任意邻近作用均与标准作用半共轭。
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with …
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.
Topological stability of relatively hyperbolic groups acting on their boundaries
本文证明了双曲群在其Bowditch边界上作用的拓扑稳定性，指出在抛物子群上受半共轭控制的小扰动必全局半共轭于标准作用，从而确立了arXiv:2206.14914主要结果的相对版本。
We consider the class of countable groups possessing an action on a finite product of hyperbolic graphs where every infinite order element acts loxodromically. When the graphs are locally finite, we obtain strong structure theorems for the groups in this subclass, so that mapping class groups of genus at least 3 (and $Aut(F_n)$ and $Out(F_n)$ for $n\geq 4$) are not in this subclass. This contrasts with the general case, where Bestvina, Bromberg and Fujiwara showed the existence of proper actions of mapping class groups on a finite product of quasitrees. In particular these quasitrees cannot be…
We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.
In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space.
Stability of hyperbolic groups acting on their boundaries
本文利用边界点的动力学编码，证明双曲群在其 Gromov 边界上的作用具有拓扑稳定性，并为圆边界情形的全局稳定性提供新证明。
We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space on which it acts as a discrete group of translations with the rhombic dodecahedron as a fundamental domain. An abundance of actions of the second kind is described. Constraints on maps from…
Recognition and constructive membership for purely hyperbolic groups acting on trees
提出基于 Nielsen 约化方法的算法，识别树自同构群是否纯双曲并给出自由基，解决构造性成员问题，附 Magma 实现与性能报告。
The goal of this article is to survey some recent developments in the study of groups acting on hyperbolic spaces. We focus on the class of acylindrically hyperbolic groups; it is broad enough to include many examples of interest, yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context. In particular, we discuss group theoretic Dehn filling and small cancellation theory in acylindrically hyperbolic groups. Many results discussed here rely on the new generalization of relative hyperbolicity based on the notion of a hyperbolically em…
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…
In their first article, the authors initiated a systematic study of hyperbolic $Λ$-metric spaces, where $Λ$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $Λ= \mathbf{Z}^n$ taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic $\mathbf{Z}^n$-metric spaces. Under certain constraints, the structure of such groups is described in terms of a {\em hierarchy} similar to the one established for $\mathbf{Z}^n$-free groups by Kharlampovich, Myasnikov, Remeslennikov and Serbin.
Drift and joint minimal displacement for groups acting on hyperbolic spaces
该研究针对作用于双曲空间的群，基于联合极小位移得到随机游走漂移的统一下界，其主定理适用于弱轴化作用，还在相关群的漂移谱中得到零处的统一间隙。
In this article, we construct partial periodic quotients of groups which have a non-elementary acylindrical action on a hyperbolic space. In particular, we provide infinite quotients of mapping class groups where a fixed power of every pseudo-Anosov homeomorphism is identified with a periodic or reducible element.
We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a group acting essentially and non-elementarily on a finite dimensional irreducible CAT(0) cube complex is acylindrically hyperbolic if there exist two hyperplanes whose stabilisers intersect along a finite subgroup. We also give further conditions on the geometry of the complex so that the result holds if we only require the existence of a single pair of points whose stabilisers intersect along a fin…
We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter.
In our paper arXiv:1310.6289, we stated that acylindrical hyperbolicity of a group is invariant under commensurability up to finite kernels. Unfortunately, the proof of this fact contained a gap. The goal of this erratum is to point out the gap and to correct the statements and proofs of results of arXiv:1310.6289 and the subsequent paper arXiv:1501.03066 affected by it.
A group with a geometric action on some hyperbolic space is necessarily word hyperbolic, but on the other hand every countable group acts (metrically) properly by isometries on a locally finite hyperbolic graph. In this paper we consider what happens when a group acts isometrically on a restricted class of hyperbolic spaces, for instance quasitrees. We obtain strong conclusions on the group structure if the action has a locally finite orbit, especially if the group is finitely generated. We also look at group actions on finite products of quasitrees, where our actions may be by automorphisms o…
Hyperbolic groups acting improperly
本文研究双曲群在 CAT(0) 立方体复形上的非真作用，通过 Sageev 构造建立超平面与胞腔稳定子拟凸性的联系，并推广 Agol 与 Wise 的相关定理。
Groups Acting Acylindrically on Trees
本文研究了在树上实现圆柱形且非元素作用的群，分类了图积群和某些双曲 3-流形的基本群，并提供了具有树作用但无非元素圆柱形树作用的圆柱形双曲群的例子。
Dynamics of primitive elements under group actions
本文研究部分本原元固定点的群作用，借助动力学工具提出 Nielsen 围长概念，并证明存在围长无限而 Nielsen 围长有限的群。
In this paper we study group actions on hyperbolic $Λ$-metric spaces, where $Λ$ is an ordered abelian group. $Λ$-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicty for such spaces. Only the case of 0-hyperbolic $Λ$-metric spaces (that is, $Λ$-trees) was systematically studied, while the theory of general hyperbolic $Λ$-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from…
We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification shows 23 non-isomorphic torsion free groups (obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the Euclidean case, we find both torsion and torsion free groups acting on the same buildi…
We prove that for a one-ended hyperbolic graph $X$, the size of the quotient $X/G$ by a group $G$ acting freely and cocompactly bounds from below the number of simplices in an Eilenberg-MacLane space for $G$. We apply this theorem to show that one-ended hyperbolic cubulated groups (or more generally, one-ended hyperbolic groups with globally stable cylinders à la Rips-Sela) cannot contain isomorphic finite-index subgroups of different indices.
In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups. We determine when the kernel of such a representation is torsion-free and thus the quotient a hyperbolic n-manifold.
Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix any point of $\partial S$. We prove that every invariant random subgroup of $G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $(X,μ)$ either has finite stabilizers…
Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups
本文证明单射度量空间上均匀适当作用群的无限循环子群均匀无畸变，并应用于层级双曲群研究平移长度，用拟态射构造最优性例子。
We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT($0$) spaces. We prove that a group acting on an irreducible non-spherical non-affine building is acylindrically hyperbolic provided there is a chamber with finite stabiliser whose orbit contains an apartment. Finally, we show that the following classes of groups admit an action on a building with those properties: orthogonal forms of Kac--Moody groups over arbitrary fields, and irreducible graph products of arbitrary groups - recov…
This paper shows that every Gromov hyperbolic group can be described by a finite subdivision rule acting on the 3-sphere. This gives a boundary-like sequence of increasingly refined finite cell complexes which carry all quasi-isometry information about the group. This extends a result from Cannon and Swenson in 1998 that hyperbolic groups can be described by a recursive sequence of overlapping coverings by possibly wild sets, and demonstrates the existence of non-cubulated groups that can be represented by subdivision rules.
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…
Convex cocompact groups with three-dimensional limit sets
本文提出一种构造三维极限集的凸紧致超反射群的方法，通过输入三维 simplicial 复形生成超反射群，并证明其为薄子群。
Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces
该研究证明了每个有限群都可以作为具有 (T) 性质的超限群的外自同构群，并展示了如何构造具有特定外自同构群的超限群。
Cubulated hyperbolic groups admit Anosov representations
本文证明了立方超几何群允许射影Anosov表示，并展示了超几何准凸子群在右角Coxeter群中的表示性质。
Subgroup mixing and random walks in groups acting on hyperbolic spaces
研究柱面双曲群通过共轭作用于无限指数凸紧子群空间的拓扑动力学，利用合适概率测度μ的随机游走产生强混合元素，证明相关技术结果并扩展前人随机游走成果。
Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups
本研究证明满足条件的群图基本群自同构可用Bestvina-Handel型不可约轨道轨形映射表示，推导自由积自同构范式并证明其非拟测地，还证实有限生成字双曲群外自同构满足推广Nielsen-Thurston分类的动力学三分性。
Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of 1-dimensional connected boundaries. We get that any non-torsion infinite f.g. group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually $\mathbb{Z} + \mathbb{Z}$).
A general construction of simultaneously hyperbolic elements
本文在最弱条件下显式构造了作用于有限多个Gromov双曲空间的群中的同时双曲元素，并证明其集合具有严格正密度，推广了多个已知结果。
Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, where E(G) is the unique maximal finite normal subgroup of G.
Sublinear projection tracking in acylindrically hyperbolic groups
研究非圆柱双曲群字度量中的投影现象，对斜驶WPD元素证明相关最短投影性质，借此得到增长函数上界、构造真商群，还证明了受限子群的增长 - 余增长不等式。
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a h…
Horocyclic products have Y-posets of hyperbolic structures
本文证明，在环游乘积上几何且正向作用的群的所有双曲结构均来自因子上的作用，并给出了第一个在两种混合千层空间的环游乘积上几何作用的群例子。
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the appropriate sense, to elements of P. As a special case of this result is a new proof, in the cubical case, of a Theorem of Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0) spaces. Second, we relate the existence of cut-points in asymptotic cones of a cub…
We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance $π$ apart are the endpoints of a geodesic in the 2-comp…
In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen, and that under certain conditions a group action $G \curvearrowright X$ induces an acylindrical action $G \curvearrowright \mathcal{C}X$, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto. As an application, we exhibit an…
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 $\tilde A_n$'' …
Uniform exponential growth for groups with proper product actions on hyperbolic spaces
本文研究有限生成群在双曲空间有限乘积上的正常作用，证明在粗稠密轨道或阴影性质下非虚阿贝尔子群具有均匀指数增长，并在弱非柱状条件下给出乘积集增长的完整分类。
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups that are also an arithmetic lattice in PSL_2(R). We consider similar finiteness theorem for complex hyperbolic triangle groups. In particular, we show that there are finitely many complex hyperbolic triangle groups with rational angular invariant which determine an arithmetic latt…
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…
In this article we study surface subgroups of groups acting simply transitively on vertex sets of certain hyperbolic triangular buildings. Kangaslampi and Vdovina have constructed and classified all groups acting simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. These groups gave the first examples of cocompact lattices acting simply transitively on vertices of hyperbolic triangular Kac-Moody buildings that are not right-angled. The hyperbolic buildings arise as universal covers of polyhedra glued together from with 15 triangular face…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。