François Dahmani, David Futer, Daniel T. Wise
We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel, and Osin on rotating families.
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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.
Ascending Chains of Free Quasiconvex Subgroups
本文证明了双曲群中不存在常秩自由拟凸子群的严格上升链，这一结果深化了对双曲群子群结构的理解。
In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.
We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are a…
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex s…
We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a question posed by D. Osin \cite{Os06}. In the second part of the paper we prove that a subgroup $H$ of a finitely generated relatively hyperbolic group $G$ acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every its infinite intersection with a pa…
A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups
本文提出层次超几何群子群生成 amalgamated 自由积的条件，并探讨其在映射类群几何子群中的应用及凸性保持问题。
We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with the graph product provided all of its quasiconvex subgroups are separable. We obtain a similar result for uniform lattices of the Davis complex of Gromov-hyperbolic two-dimensional Coxeter groups. We also prove that every extension of a uniform lattice of a CAT(0) square complex…
Consider a hyperbolic group G and a quasiconvex subgroup H of infinite index. We construct a set-theoretic section s of the quotient map (of sets) from G to G/H such that s(G/H) is a net in G; that is, any element of G is a bounded distance from s(G/H). This section arises naturally as a set of points minimizing word-length in each fixed coset gH. The left action of G on G/H induces an action on s(G/H), which we use to prove that H contains no infinite subgroups normal in G.
On the existence of weakly malnormal quasiconvex subgroups of hyperbolic groups
本文证明了任意非初等双曲群中弱法正规、几乎自由、拟凸子群的存在性，推广了Ilya Kapovich关于无挠双曲群中法正规拟凸子群的结果。
In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite height if and only if $H$ is strongly quasiconvex. On the other hand, if $M$ has a summand supporting the Sol geometry in its sphere-disc decomposition then $π_1(M)$ contains finitely generated, finite height subgroups which are not strongly quasiconvex. We also characterize stro…
For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran and Genevois. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. …
We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgro…
We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them. We also…
An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups. However, not much is yet learned about the structure of arbitrary quasiconvex subsets. In this work we study the properties of products of quasiconvex subgroups; we show that such sets are quasiconvex, their finite intersections have a similar algebraic representation and, thus, are quasiconvex too.
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.
Let $1 \to K \longrightarrow G \stackrelπ\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=π^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed surface group or a free group and $Q$ is convex cocompact), we show that infinite index quasiconvex subgroups of $G_1$ are quasiconvex in $G$. Related results are proven for metric bundles, developable complexes of groups, and graphs of groups.
In this paper we analyze the action of a quasiconvex subgroup of F_m x Z^n on the convex hull of its orbit and we show that this action is cocompact. Further, using our techniques, we obtain complete description of the quasiconvex subgroups of F_m x Z^n.
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.
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.
This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral structure of $(G,\mathbb{P})$ on $H$.
We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup $H$ of a word hyperbolic group $G$ such that the limit set of $H$ is not the limit set of a quasiconvex subgroup of $G$. In particular, this gives a counterexample to the conjecture of G.Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer.
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.
On intersections of locally quasiconvex subgroups
该研究针对双曲群的无挠局部拟凸子群，证明了其交的秩可由原二子群的秩界定，补充了双曲群子群交的秩相关结论。
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.
Given a graph of groups $\mathcal{G} = (Γ, \{G_v\}, \{G_e\})$ with certain conditions on vertex groups and $G$ acts acylindrically on its Bass-Serre tree $T$. Let $H$ be a finitely generated subgroup of $G$. We prove the following statements equivalence: $H$ has finite height, $(G, T, H)$ is a $A/QI$--triple, $H$ is strongly quasiconvex and virtually free in $G$. We also give a condition to determine whether strong quasiconvexity in a group is preserved under amalgams.
We consider two families of subgroups of a group. Each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is supposed to be hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.
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。
We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.
We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how the results of [2, Appendix A] can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.
In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundary.
Structure of quasiconvex virtual joins
本文研究了相对双曲群中相对拟凸子群的虚拟并结构，证明了此类并与极大抛物子群的交集可分解为因子子群与极大抛物子群交集的并，并为抛物子群几乎相容的拟凸子群提供了组合定理。
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…
The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-cocompact action by homeomorphisms on a compactum is finitely generated with respect to a system of parabolic subgroups. The second part essentially uses the methods of topological entourages developed in the first part. Using also Floyd metrics we obtain finer properties of finitel…
A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results w…
Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when quasicovexity relative to A implies quasiconvexity relative to B. We also show that quasiconvexity relative to B implies quasiconvexity relative to A. Some applications are discussed.
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminat…
A Kleinian group $Γ< \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $Γ$ in $\mathbb H^3$ is quasiconvex or, equivalently, $Γ$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent…
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.
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。
Let $Γ$ be a simplicial, finite, connected graph such that $Γ$ does not decompose as a nontrivial join. We prove that two notions of strong quasiconvexity and stability are equivalent in the right-angled Artin group $A_Γ$ (except for the case of finite index subgroups). We also characterize non-trivial strongly quasiconvex subgroups of infinite index in $A_Γ$ (i.e. non-trivial stable subgroups in $A_Γ$) by quadratic lower relative divergence. These results strengthen the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.
We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian groups have property (Q). We also give some applications of our results to one-relator groups and exponential groups.
Deciding if a hyperbolic group splits over a given quasiconvex subgroup
本文提出判定双曲群沿给定拟凸子群是否分裂的算法，并扩展至计算过滤端数与Schreier图端数，利用带标号有向图编码边界连通性信息。
$H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his paper in 1993 A. Ol'shanskii gave a description of all $G$-subgroups in any given non-elementary hyperbolic group $G$. Here we show that for the same class of $G$-subgroups the finiteness assumption on $M$ (under certain natural conditions) can be replaced by an assumption of quasiconvexity.
A group $G$ is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if $G$ is a hyperbolic group, $H_1$ is a quasiconvex subgroup of $G$, and $H_2$ is a subgroup of $G$ which is elementwise conjugate into $H_1$, then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellat…
In this paper, we prove that all finitely generated malnormal subgroups of one-ended right-angled Coxeter groups are strongly quasiconvex and they are in particular quasiconvex when the ambient groups are hyperbolic. The key idea is to prove all infinite proper malnormal subgroups of one-ended right-angled Coxeter groups are join-free and then prove the strong quasiconvexity and the virtual freeness of these subgroups. We also study the subgroup divergence of join-free subgroups in right-angled Coxeter groups and compare them with the analogous subgroups in right-angled Artin groups. We charac…
The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include…
We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.
For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not relatively quasiconvex in $G^*$. This generalizes a result of I. Kapovich for hyperbolic groups. We also prove that any torsion-free group $G$ that is non-elementary and hyperbolic relative to $\mathbb P$, contains a rank 2 free subgroup $F$ such that the group generated by "randomly" chosen elements $r_1,...,r_m$ in $F$ is aparabolic, malnormal in $G$ and quasic…
The main result of this paper states that for any group $G$ with an automatic structure $L$ with unique representatives one can construct a uniform partial algorithm which detects $L$-rational subgroups and gives their preimages in $L$. This provides a practical, not just theoretical, procedure for solving the occurrence problem for such subgroups.
We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.
For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensional Poincare Duality group $PD(2k+1)$. This generalizes (2). We…
We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb-Mosher and Farb.
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…
A random group contains many quasiconvex surface subgroups.
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.
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $Out(F_n)$. Corrigendum: there is an unfortunate mistake in the statement and the proof of Proposition 5.1. This affects one direction of the implications of the main theorem. A correction is given, that states that given a quasi-convex subgroup of a hyperbolic (or relatively hyperbolic) group, the graded relative hy…
We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.
A word hyperbolic group $G$ is called GFERF if every quasiconvex subgroup coincides with the intersection of finite index subgroups containing it. We show that in any such group, the product of finitely many quasiconvex subgroups is closed in the profinite topology on $G$.
Hierarchies for Relatively Hyperbolic Virtually Special Groups
本文为相对超几何虚拟紧致特殊群构造了新的虚拟准凸分层，并推广了Wise的Malnormal特殊商定理。
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