Cornelia Drutu, Mark Sapir
We prove that a finitely generated group $G$ hyperbolic relative to the collection of finitely generated subgroups H_1,..., H_m has the Rapid Decay property if and only if each H_i, i=1,2,..., m, has the Rapid Decay property.
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We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb's and Gromov's definitions.
We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.
We prove that a non-elementary relatively hyperbolic group is statistically hyperbolic with respect to every finite generating set. We also establish statistical hyperbolicity for certain direct products of two groups, one of which is relatively hyperbolic.
We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.
We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group.
The asymptotic bound for a length-based attack on the Conjugacy Search Problem in relatively hyperbolic groups is cubic for hyperbolic elements and a "small" polynomial for parabolic elements, depending on the Conjugacy Search Problem for the peripheral subgroups. The bound for relatively hyperbolic groups in this paper is a significant improvement on previous work.
In this paper we give new requirements that a tree of $δ$-hyperbolic spaces has to satisfy in order to be $δ$-hyperbolic itself. As an application, we give a simple proof that limit groups are relatively hyperbolic.
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.
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 $G$.
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.
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.
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.
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.
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.
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…
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。
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.
The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type $HF^\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of su…
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…
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.
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.
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.
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 (…
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.
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.
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…
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.
We prove that the topological complexity $\mathrm{TC}(π)$ equals $\mathrm{cd}(π\timesπ)$ for certain toral relatively hyperbolic groups $π$.
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…
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.
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.
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。
The purpose of this note is to provide a short alternate proof that (combined with a theorem proven by Szczepanski) shows that a group which is relatively hyperbolic in the sense of the definition of Gromov is relatively hyperbolic in the sense of the definition of Farb.
We characterize relatively hyperbolic groups whose reduced $C^*$-algebra is simple as those, which have no non-trivial finite normal subgroups.
We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. (1) If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. (2) If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic group is not Hopfian, then H acts non-trivially on an R-tree. (3) Every non-elementary relatively hyperbolic group has a non-elementary relative…
Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.
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.
Bi-exactness of relatively hyperbolic groups
本研究证明有限生成相对双曲群具有双精确性的充要条件是其所有外围子群均为双精确的，推广了Ozawa关于外围子群顺从时相对双曲群双精确的结论。
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.
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.
We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperb…
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.
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.
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.
Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups.
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。
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。
We prove that, if a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, by use of M. Bonk and O. Schramm embedding theorem, a very short proof of the finiteness of asymptotic dimension of relatively hyperbolic groups with virtually nilpotent parabolic subgroups (which is known to imply Novikov conjectures
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.
In this paper we prove that if some relatively hyperbolic groups have Bowditch boundary homeomorphic to the $n$-sphere, then they are also relatively hyperbolic with respect to another set of parabolic subgroups and its Bowditch boundary is homeomorphic to the $n-1$-dimensional Sierpiński carpet.
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…
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…
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…
We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the divergence of (properly) relatively hyperbolic groups, in particular showing that it is at least exponential. Our main tool is the generalization of a result proved by Bowditch for hyperbolic spaces: if a family of paths in a space satisfies a list of properties specific to geodes…
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.
We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H depending on the relative quasiconvexity of the intersection of H with the vertex groups of G. We give an application towards local relative quasiconvexity.
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2πTheorem in the …
In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.
Baker and Riley proved that a free group of rank 3 can be contained in a hyperbolic group as a subgroup for which the Cannon-Thurston map is not well-defined. By using their result, we show that the phenomenon occurs for not only a free group of rank 3 but also every non-elementary hyperbolic group. In fact it is shown that a similar phenomenon occurs for every non-elementary relatively hyperbolic group.
Parabolic cut pairs in boundaries of relatively hyperbolic groups
本文针对相对双曲群边界中未被探索的抛物切割对现象，提出组合定理构造相关群示例并证明其完备性，同时给出两种适用的边界拓扑描述。
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