Growth of relatively hyperbolic groups

Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.

Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $ϕ$ into a relatively hyperbolic group $G$ is relatively hyperbolic with respect to a system of subgroups whose image under $ϕ$ is situated in a uniformly bounded distance from the parabolic subgroups of $G$.

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.

A Whitehead algorithm for toral relatively hyperbolic groups

We prove that there is an algorithm to determine whether a tuple of elements in a toral relatively hyperbolic group G is in the automorphic orbit of the other tuple.

On the quasiconvexity of the fixed subgroup of endomorphisms of relatively hyperbolic groups

We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.

An Algorithm to Solve the Generalized Conjugacy Problem in Relatively Hyperbolic Groups

In this note, we provide a (super-exponential time) algorithm to solve the generalized conjugacy problem in relatively hyperbolic groups, given solvability of the generalized conjugacy problem in each of the parabolic subgroups.

Relatively hyperbolic metric bundles and Cannon-Thurston map

Given a metric (graph) bundle $X$ over $B$ where all the fibres are strongly relatively hyperbolic and nonelementary we show that, under certain conditions, $X$ is strongly hyperbolic relative to a collection of maximal cone-subbundles of horosphere-like spaces. Further, given a coarsely Lipschitz qi embedding $i: A\to B$, we show that the pullback $Y$ is strongly relatively hyperbolic and the map $Y\to X$ admits a Cannon-Thurston (CT) map. As an application, we prove a group-theoretic analogue of this result for a relatively hyperbolic extension of groups.

Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures…

相对超几何群中的折叠

Foldings in relatively hyperbolic groups

研究了相对超几何群中子群的载体图，证明了相对准凸子群的组合定理，并引入了折叠理论，得出局部相对准凸的相对超几何群和克莱因群子群的有限性结果。

The compressed word problem in relatively hyperbolic groups

We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups, in two different contexts. First, we elaborate on a remark from math.GR/0601311, which states that the version of Dehn filling in relatively hyperbolic groups proved in math.GR/0510195, allowing infinitely generated parabolic subgroups, follows from the version with finitely generated parabolics. Second, we observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We use this (and known results) to derive some consequ…

Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.

The SQ-universality and residual properties of relatively hyperbolic groups

We study residual properties of relatively hyperbolic groups. In particular, we show that if a group $G$ is non-elementary and hyperbolic relative to a collection of proper subgroups, then $G$ is SQ-universal.

Accidental parabolics and relatively hyperbolic groups

By constructing, in the relative case, objects analoguous to Rips and Sela's canonical representatives, we prove that the set of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite, up to conjugacy.

A combination theorem for relatively hyperbolic groups and finite relative height of splitting

In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of graph of groups. Suppose G(Y) is a graph of relatively hyperbolic groups such that edge groups are relatively quasi-convex in adjacent vertex groups. Also, assume that the fundamental group of G(Y) is relatively hyperbolic. Then we show that the edge groups of G(Y) have finite relative height (Definition 1.5) if and only if they are relatively quasi-convex (…

A Cartan-Hadamard type result for relatively hyperbolic groups

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of M. Sapir.

On the generalized membership problem in relatively hyperbolic groups

The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on the peripheral structure of these groups. These hypotheses are satisfied, in particular, by toral relatively hyperbolic groups.

Specializing cubulated relatively hyperbolic groups

In arXiv:1204.2810 Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof proceeds by first recubulating to obtain an improper action with controlled stabilizers (a weakly relatively geometric action), and then Dehn filling to obtain many cubulated hyperbolic quotients. We apply our results to prove the Relative Cannon Conjecture for certain cubulated or parti…

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.

Time complexity of the conjugacy problem in relatively hyperbolic groups

If $u$ and $v$ are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok. Bridson and Haefliger showed that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in pol…

Combination of quasiconvex subgroups of relatively hyperbolic groups

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups $Q$ and $R$ is quasiconvex and isomorphic to $Q \ast_{Q\cap R} R$. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented.

Symbolic dynamics and relatively hyperbolic groups

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this condition is satisfied, including geometrically finite kleinian groups.

相对双曲TDLC群的拟等距不变性

Quasiisometry invariance of relatively hyperbolic TDLC groups

本文引入仅依赖Cayley-Abels图的相对双曲TDLC群新定义，证明其与既有定义等价，并证明该性质在拟等距下不变。

Relative Quasiconvexity using Fine Hyperbolic Graphs

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic.

Asymptotic dimension of relatively hyperbolic groups

Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$ is also finite.

A note on the Farrell-Jones conjecture for relatively hyperbolic groups

For a group G relatively hyperbolic to a family of residually finite groups satisfying the Farrell-Jones conjecture, we reduce the solution of the Farrell-Jones conjecture for G to the case of certain nice cyclic extensions in G.

Peripheral structures of relatively hyperbolic groups

In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using dynamical methods. Some applications are discussed. The class of groups acting geometrically finitely on Floyd boundaries turns out to be easily understood. However, we also show that Dunwoody's inaccessible group does not act geometrically finitely on its Floyd boundary.

Detecting free splittings in relatively hyperbolic groups

We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.

关于一个 Hopf 子群的非 Hopf 相对双曲群

A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup

本文通过构造一个相对于 Hopf 子群的非 Hopf 有限生成相对双曲群反例，否定回答了 Osin 提出的问题5.5。

具有强捷径抛物子群的相对双曲群是强捷径群

Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut

本文证明了相对于强捷径群为双曲的群本身也是强捷径群，获得了该群的新例子，并证明了相对双曲群及其抛物子群在图上的余紧作用性质。

Small cancellations over relatively hyperbolic groups and embedding theorems

We generalize the small cancellation theory over hyperbolic groups developed by Olshanskii to the case of relatively hyperbolic groups. This allows us to construct infinite finitely generated groups with exactly $n$ conjugacy classes for every $n\ge 2$. In particular, we give the affirmative answer to the well--known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.

The Isomorphism Problem for Toral Relatively Hyperbolic Groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela); and (ii) fully residually free groups (Bumagin, Kharlampovich and Miasnikov). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for $n \ge 3$. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabo…

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…

Fixed subgroups of automorphisms of relatively hyperbolic groups

Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this hap…

Negligibility of parabolic elements in relatively hyperbolic groups

We study density of parabolic elements in a finitely generated relatively hyperbolic group $G$ with respect to a word metric. We prove this density to be zero (apart from degenerate cases) and the limit defining the density to converge exponentially fast; this has recently been proven independently by W. Yang. As a corollary, we obtain the analogous result for the set of commuting pairs of elements in $G^2$, showing that the degree of commutativity of $G$ is equal to zero.

Extending Properties to Relatively Hyperbolic Groups

Consider a finitely generated group $G$ that is relatively hyperbolic with respect to a family of subgroups $H_1, ..., H_n$. We present an axiomatic approach to the problem of extending metric properties from the subgroups $H_i$ to the full group $G$. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.

构造具有 Hopfian 边缘子群的 non-Hopfian 相对双曲群的方法

A recipe for constructing non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups

本文利用像扩张定理提出了一种构造具有 Hopfian 边缘子群的 non-Hopfian 相对双曲群的一般方法，并给出了两个具体实例，为 Osin 问题提供了反例。

相对双曲的实际上特殊群的profinite完备化

The profinite completion of relatively hyperbolic virtually special groups

本文利用profinite完备化刻画toral相对双曲的实际上特殊群，并证明其完备化子群的Tits二择一与有限生成pro-p子群结构，结果应用于双曲算术流形及同余核。

Elementary subgroups of relatively hyperbolic groups and bounded generation

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…

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite subsystem, and a number of structural results about Gamma-limit groups.

Permute and conjugate: the conjugacy problem in relatively hyperbolic groups

Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…

一个非一致指数增长的圆柱双曲群

An acylindrically hyperbolic group of non-uniform exponential growth

构造了一个具有非一致指数增长和Kazhdan性质(T)的圆柱双曲群，该群是递增有限秩自由群的自同构群的公共商。

Growth of periodic quotients of hyperbolic groups

Let G be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient G/G^n tends to the one of G as n odd approaches infinity. Moreover we provide an estimate at which the convergence is taking place.

关于相对双曲TDLC群的组合定理与Bowditch边界

On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

本文为全不连通局部紧群引入相对双曲性概念，证明其与Arora-Pedroza定义等价，并建立融合积与HNN扩张的组合定理，构造Bowditch边界，进而刻画粗糙端无限时边界的唯一确定性和单端时的连通性，最后给出Gromov边界全不连通时群分裂为紧群有限图的结论。

Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of $(W,S)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $(W,S)$ are isolated.

在群复形上粘合相对双曲群

Combining relatively hyperbolic groups over a complex of groups

本研究针对满足特定条件的相对双曲群构成的群复形，通过构造其基本群的Bowditch边界模型并结合Yaman定理，证明该基本群是相对双曲的，还给出Bowditch边界的一般分解方法，为相关拓扑问题研究提供工具。

相对双曲对的相对上同调维数

Relative cohomological dimension of a relatively hyperbolic pair

该研究证明无挠相对双曲对的相对上同调维数有限，且在特定条件下拟等距不变，并通过所提方法计算了多类情形的该维数。

相对双曲群中虚拟联接的拟凸性与乘积的可分性

Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

本文证明了在QCERF相对双曲群中，有限生成相对拟凸子群存在有限指数子群使虚拟联接保持拟凸性，并由此导出多类群中子群乘积的可分性。

Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces

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.

Packing subgroups in relatively hyperbolic groups

We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, und…

Boundaries of graphs of relatively hyperbolic groups with cyclic edge groups

We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups with dynamically quasi-convex edge groups (Theorem 1.3). To prove these results, we use a modification of Dahmani's technique [Dah03]. Then we show that the fundamental group of a graph of relatively hyperbolic groups with edge groups either parabolic or infinite cyclic is relatively hyperbolic and construct Bowditch boundary. Finally, we show that the homeo…

Formal conjugacy growth in acylindrically hyperbolic groups

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely gene…

有限性性质与相对双曲群

Finiteness properties and Relatively Hyperbolic Groups

本文证明了相对双曲群的有限性性质 $F_n$ 和 $FP_n$ 成立当且仅当其所有边缘子群成立，并以此构造出可数无穷多类具有特定有限性性质差异的单头群拟等距类。

将相对双曲自同构的共轭问题归约为外围分量问题

Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

本研究将自由群与无挠双曲群的外自同构共轭问题归约为多项式增长自同构映射环面的相关算法问题，求解了多类新外自同构的共轭问题，为彻底解决$Out(F_n)$的共轭问题提供了可行路径。