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 characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.
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.
For noncompact semisimple Lie groups $G$ we study the dynamics of the actions of their discrete subgroups $Γ<G$ on the associated partial flag manifolds $G/P$. Our study is based on the observation that they exhibit also in higher rank a certain form of convergence type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the $Γ$-action on various domains of proper discontinuity, in particular on domains in the full flag manifold $G/B$…
We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).
We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular, such subgroups are quasiconvex in $A(Γ)$. In addition, we identify a milder condition for a finitely generated subgroup of $A(Γ)$ that guarantees it is free, undistorted, and retains finite generation when intersected with $A(Λ)$ for subgraphs $Λ$ of $Γ$. These results have applications to both the study of convex cocompactness in $\text{Mod}(S)$ and the way…
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 study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to conve…
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.
Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup H<G is closed in the pro-normal topology. As a corollary we deduce that if M is a maximal subgroup of a lattice in PSL(2,C) then either M is finite index or M is not finitely generated.
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…
It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
We prove that the free product of two finitely presented locally tame groups is locally tame and describe many examples of tame subgroups of finitely presented groups. We also include some open problems related to tame subgroups.
It is proved that all finitely generated subgroups of generalized free product of two groups are finitely separable provided that free factors have this property and amalgamated subgroups are normal in corresponding factors and satisfy the maximum condition for subgroups.
We prove that for all $k,m,n \in \mathbb N \cup \{\infty\}$ with $4 \leq k \leq m \leq n$, there exists a finitely generated group $G$ with a finitely generated subgroup $H$ such that the asymptotic dimension of $G$ is $k$, the Assouad-Nagata dimension of $G$ is $m$, and the Assouad-Nagata dimension of $H$ is $n$. This simultaneously answers two open questions in asymptotic dimension theory.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $Γ(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $ΔP_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.
In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is closely related to the separability of this subgroup.
A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.
There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittlesey's group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of …
Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine completely when this subgroup is finitely generated, by using preceding results on centralizers of reflections. This result provides a new and naturally constructed example of a family of non-finitely generated subgroups of finitely generated groups.
We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary of this theorem says that a noncyclic free group of finite rank admits a faithful highly transitive action on an infinite set, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.
Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings.
We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil\frac{n}{3}\rceil$.
We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff the Hausdorff distance between A and gA is finite. We show that a quasi-normal subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is quasi-normal iff the natural coset graph is locally finite. This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary go…
We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.
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 give a short proof of a theorem of Handel and Mosher stating that any finitely generated subgroup of $\text{Out}(F_N)$ either contains a fully irreducible automorphism, or virtually fixes the conjugacy class of a proper free factor of $F_N$, and we extend their result to non finitely generated subgroups of $\text{Out}(F_N)$.
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 …
For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all fin…
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results e…
We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.
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.
For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…
We show that a nonempty family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $Γ$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \neq 1$ is the greatest subgroup of $Γ$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the an…
Let G be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup H of Aut(G) consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.
We study 3-dimensional Poincaré duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of $G$. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-$p$ groups.
We prove that if a surface group embeds as a normal subgroup in a Kähler group and the conjugation action of the Kähler group on the surface group preserves the conjugacy class of a non-trivial element, then the Kähler group is virtually given by a direct product, where one factor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infinite outer automorphism group embeds as a normal subgroup in a Kähler group then it is virtually a surface group. More generally we give restrictions on normal subgroups of Kähler groups which are amalgamated products or HNN extension…
We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain semistability condition, which generalizes the notion of stability introduced by…
We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.
Consider the mapping class group $\Mod_{g,p}$ of a surface $Σ_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection $\{f_1,...,f_k\}$ of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large $N$, the mapping classes $\{f_1^N,...,f_k^N\}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When $\{f_1,...,f_k\}$ are a…
We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.
We prove that if $G$ is a finitely generated group and $Z$ is a uniformly recurrent subgroup of $G$ then there exists a minimal system $(X,G)$ with $Z$ as its stability system. This answers a query of Glasner and Weiss \cite{GW} in the case of finitely generated groups. Using the same method (introduced by Alon, Grytczuk, Haluszczak and Riordan \cite{AGHR}) we will prove that finitely generated sofic groups have free Bernoulli-subshifts admitting an invariant probability measure.
We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as an investigation into the difference between strong and weak effective coherence for finitely presented groups.
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group.
We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for …
A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …
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…
A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.
We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than $3$.
We prove a variant of the well-known Reidemeister-Schreier theorem for finitely $L$-presented groups. More precisely, we prove that each finite index subgroup of a finitely $L$-presented group is itself finitely $L$-presented. Our proof is constructive and it yields a finite $L$-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely $L$-presented group to be invariantly $L$-presented itself.
