If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.
We prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group is ERF iff it is polycyclic.
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 study ascending HNN-extensions $G$ of finitely generated free abelian groups: examples of such $G$ include soluble Baumslag-Solitar groups and fundamental groups of orientable prime $3$-manifolds modelled on Sol geometry. In particular, we study the elliptic subgroup $A \leq G$, consisting of all elements that stabilise a point in the Bass-Serre tree of $G$. We consider the density of $A$ with respect to ball counting measures corresponding to finite generating sets of $G$, and we show that $A$ is exponentially negligible in $G$ with respect to such sequences of measures. As a consequence, …
A group $G$ is said to have the Howson property (or to be a Howson group) if the intersection of any two finitely generated subgroups of $G$ is finitely generated subgroup. It is proved that descending HNN-extension is not a Howson group under some assumptions satisfied by the base group of HNN-extension. In particular, a result of the paper joined with a Burns - Brunner result (received in 1979) implies that any descending HNN-extension of non-cyclic free group does not have the Howson property.
This paper is devoted to the investigation of the property of order separability for HNN extensions and free products with commutative subgroups. Particularly it was proven that HNN extension of a free group with maximal connected cyclic subgroups is 2-order separable.
Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$.
A 1-ended finitely presented group has semistable fundamental group at $\infty$ if it acts geometrically on some (equivalently any) simply connected and locally finite complex $X$ with the property that any two proper rays in $X$ are properly homotopic. If $G$ has semistable fundamental group at $\infty$ then one can unambiguously define the fundamental group at $\infty$ for $G$. The problem, asking if all finitely presented groups have semistable fundamental group at $\infty$ has been studied for over 40 years. If $G$ is an ascending HNN extension of a finitely presented group then indeed, $G…
Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "m…
Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability, which implies solvability of the word problem. In this paper we give two new characterizations of the stackable property for groups, and use these to show that every HNN extension of a stackable group is stackable. We apply this to exhibit a wide range of Dehn functions that are admitted by stackable and autostackable groups, as well as an example of a stack…
We study non-ascending HNN extensions acting on their Bass-Serre trees, and characterize C*-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not C*-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning C*-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.
We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index s…
For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-c…
Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at infinity. While many classes of groups are known to contain only semistable at infinity groups, there are only a few combination results for such groups. Our main theorem is such a result. Main Theorem. Suppose $G$ is the fundamental group of a connected reduced graph of groups, where each edge group is infinite …
We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidire…
We prove a normal form result for the groupoid of germs generated by PSL(2,C) and the exponential map. As consequences, we generalize a result of Cohen about the group of translations and powers, and prove that the subgroup of Homeo(R,+infinity) generated by the positive affine maps and the exponential map is isomorphic to a HNN-extension.
We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group $Γ$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called stron…
We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir. The main ingredient in our construction is a specific finite volume (noncompact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of its fundamental gr…
In this article we prove that an "isometric multiple HNN-extension" of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise's example of a CAT(0) non-Hopfian group is Almost convex.
The mapping torus of an endomorphism Φof a group G is the HNN-extension G*_G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.
We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.
We prove that the property to be limit aperiodic is preserved by the standard construction with groups like extension, HNN extension and free product. We also construct a non-limit aperiodic G-space.
If G is a semidirect product N by H with N normal and finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications to 3-manifolds and to cyclically presented groups, for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a cyclic cover of M whose fundamental group surjects to S. We also give a short proof of the residual finiteness of ascending HNN extensions of finite rank free groups when the induced map on homology is injective.
We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finite…
Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an edge stabilizer. Using this result we prove under certain conditions that free pro-p products with procyclic amalgamation inherit from its free factors the property of each 2-generated subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogs of classical results in abstract combinatorial group theory.
The finitely generated intersection property in fundamental groups of graphs of groups
Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）；Instituto de Ciencias Matemáticas (ICMAT)（数学科学研究所）；University of the Basque Country - EHU（巴斯克大学）；Harish-Chandra Research Institute, A CI of HBNI（哈里什-钱德拉研究所）；CNRS, ReLaX, IRL 2000, Siruseri, India（法国国家科学研究中心，ReLaX）；CNRS, Univ. Sorbonne Paris Nord, LIPN, UMR 7030, F-93430 Villetaneuse, France（法国国家科学研究中心，索邦巴黎北大学，LIPN）
It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows…
We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.
