We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of divergence, and lower relative divergence generalizes a definition of Cooper-Mihalik. While the lower divergence of Cooper-Mihalik can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this paper, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or a cyclic subgroup. We also explore relative divergence of $\CAT(0)$ groups and relatively hyperbolic groups with respect to various subgroups to better understand geometric properties of these groups.
We construct families of finitely presented groups exhibiting new divergence behavior; we obtain divergence functions of the form $r^α$ for a dense set of exponents $α\in [2,\infty)$ and $r^n\log(r)$ for integers $n \geq 2$. The same construction also yields examples of finitely presented groups which contain Morse elements that are not contracting.
We construct families of $CAT(0)$ groups containing finitely presented groups whose divergence functions are of the form $r^α$ for a dense set of exponents $α\in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.
We compute the divergence of the finitely generated group SLn(O_S), where S is a finite set of valuations of a function field, and O_S is the corresponding ring of S-integer points. As an application, we deduce that all its asymptotic cones are without cut-points.
We construct easy embeddings of relatively free groups (say the free Burnside group, the free solvable group) into finitely presented groups. We introduce a concept of verbal isoperimetric function of a group variety. We prove that if the verbal Dehn function of a relatively free group is bounded by a polynomial then the group can be embedded quasi-isometrically into a finitely presented group with polynomial isoperimetric function. We also construct an easy embedding of any Baumslag-Solitar solvable group into a finitely presented group with polynomial Dehn function.
We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.
We show the connection between the relative Dehn function of a finitely generated metabelian group and the distortion function of a corresponding subgroup in the wreath product of two free abelian groups of finite rank. Further, we show that if a finitely generated metabelian group $G$ is an extension of an abelian group by $\mathbb Z$ the relative Dehn function of $G$ is polynomially bounded. Therefore, if $G$ is finitely presented, the Dehn function is bounded above by the exponential function up to equivalence.
In this paper we describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon's completion $G^{\mathbb{Z}[t]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}[t]}$ containing $G$ is universally equivalent to $G$. Since finitely generated groups universally equivalent …
We introduce a new quasi-isometry invariant, called the divergence spectrum, to study finitely generated groups. We compare the concept of divergence spectrum with the other classical notions of divergence and we examine the divergence spectra of relatively hyperbolic groups. We show the existence of an infinite collection of right-angled Coxeter groups which all have exponential divergence but they all have different divergence spectra. We also study Morse boundaries of relatively hyperbolic groups and examine their connection with Bowditch boundaries.
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 introduce a new invariant of finitely generated groups, the ambiguity function, and prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate $\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert B^{X}_H(n) \vert}$ of a subgroup $H$ of an acylindrically hyperbolic group $G$ exists with respect to every finite generating set $X$ of $G$, if $H$ contains a loxodromic element of $G$. Further we prove that the relative exponential growth rate of every finitely generated subgro…
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.
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.
In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called "round" and "unfolded", provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer--Fleming inequality for …
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…
Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.
In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.
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…
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 main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdan's property (T) and containing exactly two conjugacy classes.
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
Suppose $G$ is a 1-ended finitely generated group that is hyperbolic relative to P a finite collection of 1-ended finitely generated subgroups. Our main theorem states that if the boundary $\partial (G, P)$ has no cut point, then $G$ has semistable fundamental group at $\infty$. Under mild conditions on $G$ and the members of P the 1-ended hypotheses and the no cut point condition can be eliminated to obtain the same semistability conclusion. We give an example that shows our main result is somewhat optimal. Finally, we improve a "double dagge" result of F. Dahmani and D. Groves.
Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and $H_2$ is not parabolic. Suppose that $H_2$ is elementwise conjugate into $H_1$. Then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. The minimal length of the conjugator can be estimated. In the case where $G$ is a limit group, it is sufficient to assume only that $H_1$ is a finitely generated and $H_2$ is an arbitrary subgroup of $…
We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of $\CAT(0)$ groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide …
Suppose $G$ is a finitely presented group that is hyperbolic relative to ${\bf P}$ a finite collection of 1-ended finitely generated proper subgroups of $G$. If $G$ and the ${\bf P}$ are 1-ended and the boundary $\partial (G,{\bf P})$ has no cut point, then $G$ was known to have semistable fundamental group at $\infty$. We consider the more general situation when $\partial (G,{\bf P})$ contains cut points. Our main theorem states that if $G$ is finitely presented and each $P\in {\bf P}$ is finitely generated and has semistable fundamental group at $\infty$, then $G$ has semistable fundamental …
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…
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.
A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn't contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups. As a corollary, slender JSJ hierarchies of hyperbolic groups which are (virtually) without $2$--torsion and finitely presented subgroups of SL(n,Z) are both fini…
For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…
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.
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.
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…
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.
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 subgroup Q is commensurated in a group G if each G conjugate of Q intersects Q in a group that has finite index in both Q and the conjugate. So commensurated subgroups are similar to normal subgroups. Semistability and simple connectivity at infinity are geometric asymptotic properties of finitely presented groups. In this paper we generalize several of the classic semistability and simple connectivity at infinity results for finitely presented groups. In particular, we show that if a finitely generated group G contains an infinite finitely generated commensurated subgroup Q of infinite inde…
We introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if $G$ is a finitely generated group acting by order preserving homeomorphism of on the line, then if some stabilizer of a point is proper and co-amenable subgroup, then $G$ surjects onto $\mathbb{Z}$. This is a generalization of a theorem of Morris.
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…
In this article, we will prove a full topological version of Popa's measurable cocycle superrigidity theorem for full shifts. More precisely, we prove that every Hölder continuous cocycle for the full shifts of every finitely generated group $G$ that has one end, undistorted elements and sub-exponential divergence function is cohomologous to a group homomorphism via a Hölder continuous transfer map if the target group is complete and admits a compatible bi-invariant metric. Using the ideas of Behrstock, Dru\c {t}u, Mosher, Mozes and Sapir, we show that the class of our acting groups is large i…
We prove that certain coinduced actions for an inclusion of finitely generated commensurated subgroups with relative one end are continuous cocycle superrigid actions. We also show the necessity for the relative end assumption.
We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf $G$-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on $\ell_{\infty}$-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.
We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic rays is asymptotically bounded away from zero. This is always satisfied by Morse geodesics. The main purpose of this paper is to furnish an example of a finitely generated group $G$ such that 1) all finitely presented subgroups of G are hyperbolic, 2) G has an uncountable family…
Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G is a toral relatively hyperbolic group and $\mathcal{A}$ is the family of abelian subgroups. We also show vertex finiteness when G is hyperbolic relative to virtually polycyclic subgroups and $\mathcal{A}$ is the family of virtually cyclic subgroups; if moreover G is one-ended, …
Let $Γ$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $Γ$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly embeddable in a Hilbert space.
The divergence of a group is a quasi-isometry invariant defined in terms of pairs of points and lengths of paths avoiding a suitable ball around the identity. In this paper we study "random divergence'', meaning the divergence at two points chosen according to independent random walks or Markov chains; the Markov chains version can be turned into a quasi-isometry invariant. We show that in many cases, such as for relatively hyperbolic groups, mapping class groups, and right-angled Artin groups, the divergence at two randomly chosen points is with high probability equivalent to the divergence o…
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.
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 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.
Let $Γ$ be a relatively hyperbolic group and let $μ$ be an admissible symmetric finitely supported probability measure on $Γ$. We extend Floyd-Ancona type inequalities up to the spectral radius of $μ$. We then show that when the parabolic subgroups are virtually abelian, the Martin boundary of the induced random walk on $Γ$ is stable in the sense of Picardello and Woess. We also define a notion of spectral degenerescence along parabolic subgroups and give a criterion for strong stability of the Martin boundary in terms of spectral degenerescence. We prove that this criterion is always satisfie…
