D. I. Moldavanskii
A criterion for the HNN-extension of a finite p-group to be residually a finite p-group is obtained and based on this criterion the sufficient condition for residuality a finite p-group of HNN-extension with arbitrary base group is proved. Then these results are applied to give for groups from two classes of one-relator groups the necessary and sufficient condition to be residually a finite p-group.
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Embedding Partial HNN Extensions In Ascending HNN Extensions
本文证明了自由群的部分升HNN扩张可嵌入实际升HNN扩张，并能作为相对双曲群的抛物子群嵌入。
We give a criterion for an HNN extension of a finite $p$-group to be residually $p$.
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.
Using recent developments on locally compact groups, we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.
Profinite genus of HNN-extensions with finite associated subgroups
本文研究了关联子群为有限群的HNN扩张的概型，给出了其同构类数量的公式并探讨了概型确定性的条件。
We show that an HNN-extension with finitely generated abelian base group is Z-linear if and only if it is residually finite.
We obtain an upper bound for relative Dehn functions of amalgamated products and HNN--extensions with respect to certain collections of subgroups. Our main results generalize the combination theorems for relatively hyperbolic groups proved by Dahmani.
On exponentiation, $p$-automata and HNN extensions of free abelian groups
本文证明对任意素数p，自由阿贝尔群的广泛HNN扩张类都能由有限p-自动机忠实表示，揭示了群论与自动机理论之间的深层联系。
The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic e…
We calculate the rank gradient and p-gradient of free products, free products with amalgamation over an amenable subgroup, and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For the p-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used.
We show that there exist finitely generated soluble groups which are not LERF but which do not contain strictly ascending HNN extensions of a cyclic group. This solves Problem 16.2 in the Kourovka notebook. We further show that there is a finitely presented soluble group which is not LERF but which does not contain a strictly ascending HNN extension of a polycyclic group.
Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asympotic cones with cut points, as well as, HNN extensions and amalgamated products where the associated subgroups are nicely embedded. We also show a slightly weaker dichotomy for multiple HNN extensions of free groups.
It is shown that the compressed word problem for an HNN-extension with base group H and finite associated subgroups is polynomial time Turing-reducible to the compressed word problem for H. An analogous result for amalgamated free products is shown as well.
We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a "new" counter-example to Gromov's conjecture on the residual finiteness of hyperbolic groups.
Given the Hilbert space compression of two groups, we find bounds on the Hilbert space compression of their free product. We also investigate the Hilbert space compression of an HNN-extension of a group relative to a finite normal subgroup or a finite index subgroup.
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.
In this paper, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Groebner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.
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.
The Conjugacy Problem for ascending HNN-extensions of free groups
该研究提出算法解决自由群上升HNN-扩张的共轭问题，同时给出相关算法处理自由群自同态动力学的特定问题，为群论相关问题的求解提供了算法支撑。
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.
We present sufficient conditions for HNN extensions to be inner amenable, respectively ICC, which give necessary and sufficient criteria among Baumslag-Solitar groups. We deduce that such a group, viewed as acting on its Bass-Serre tree, contains non trivial elements which fix unbounded subtrees.
We use the Bass--Jiang group for automorphism-induced HNN-extensions to build a framework for the construction of tractable groups with pathological outer automorphism groups. We apply this framework to a strong form of a question of Bumagin--Wise on the outer automorphism groups of finitely presented, residually finite groups.
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 find strictly ascending HNN extensions of finite rank free groups possessing a presentation 2-complex which is a non positively curved square complex. On showing these groups are word hyperbolic, we have by results of Wise and Agol that they are linear over the integers. An example is the endomorphism of the free group on a,b with inverses A,B that sends a to aBaab and b to bAbba.
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…
We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that "hard" instances of the problem comprise a negligibly small part of the group.
A non-Hopfian ascending HNN-extension of a finitely presented Hopfian group
本文通过构造一个有限生成霍普夫群的非霍普夫递增HNN扩展群，回答了关于残余有限群的HNN扩展问题。
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…
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 $y \in Y$.
Ping-pong for basis-conjugating HNN-extension of free group
本文通过基共轭嵌入的多重HNN扩张，构建了自由群的正规形式，并利用乒乓球引理发现纯辫群中的新自由子群家族。
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
Hyperbolicity of Multiple Ascending HNN Extensions of Free Groups
研究多重上升HNN扩张的超几何性，基于超几何非满射自同构的推广，分析其结构特性。
On the submonoid membership problem for HNN extensions of free groups
本文研究自由群HNN扩张中的子幺半群成员问题，提出使成员问题可判定的自然条件，并将其应用于解决此前方法无法处理的单关系逆幺半群字问题。
The intersection of the subgroups of finite $p$-index in a multiple HNN-extension of an infinite cyclic group
本文研究了无限循环群的多重 HNN 扩张中有限 $p$-指数正规子群的交集，并推广了 Moldavanskii 对 Baumslag-Solitar 群的结果。
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.
A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent $n \gg 1$ is given in the following form. Let $Q$ be a group of odd exponent $n > 2^{48}$ and $\mathcal G$ be an HNN-extension of $Q$. If $A \in \mathcal G$ then let $\mathcal F(A)$ denote the maximal subgroup of $Q$ which is normalized by $A$. By $τ_A$ denote the automorphism of $\mathcal F(A)$ which is induced by conjugation by $A$. Suppose that for every $A \in \mathcal G$, which is not conjugate to an element of $Q$, the group $<τ_A, \mathcal F(A)>$ has exponent $n$ and, in addition, equalitie…
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 show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are CAT(0) but not biautomatic. These groups also resolve a number of other questions concerning CAT(0) groups.
Examples of left-orderable and non-left-orderable HNN extensions
本文证明了 torsion-free nilpotent 群的 HNN 扩张是左可序的，并构造了非左可序的 HNN 扩张例子。
Reduced HNN extensions of von Neumann algebras (as well as $C^*$-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.
We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces
Exponent equations in HNN-extensions
本文研究有限生成群中的指数方程，针对稳定字母作用平凡的HNN-扩张，证明在额外技术条件下背包半线性性从基群转移至扩张群，覆盖中心化子、双曲群拟凸子群等场景。
On treeings arising from HNN extensions
本文为包括Baumslag-Solitar群在内的某些HNN扩张的保概率测度作用构造了商群胚的树化，并证明部分此类群的模同态核与无限秩自由群和Z的直积测度等价。
Relative hyperbolicity of ascending HNN extension of groups
该研究证明了带自由因子系统的有限生成群的保系统单自同态对应的升序HNN扩张的相对双曲性，并得出有限秩自由群指数增长单自同态的升序HNN扩张相对双曲的推论，推进了几何群论相关研究。
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, …
Stability of amalgamated free products and HNN extensions
本文考察稳定群在有限群上的合合自由积与HNN扩张，分析算子范数、Hilbert-Schmidt及置换三种稳定性，构造了多个稳定或柔性稳定的非顺从群新例。
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…
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.
Girth Alternative for HNN Extensions
本文证明有限生成群的HNN扩张子类满足周长二选一性质，并构造反例说明该性质在更广范围内一般不成立。
Certain residual properties of HNN-extensions with normal associated subgroups
本文研究了具有正规关联子群的HNN-扩张，证明了其剩余为特定群类的充分必要条件，并给出了关于剩余有限性和剩余p-有限性的新事实。
HNN extensions and embedding theorems for groups
本文探讨HNN扩张的起源、发展及其在群作用于树的Bass-Serre理论中的核心地位。
It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified subgroups.
The authors announce the following theorem. Theorem 1. If $G=A*_H B$ is an amalgamated product where $A$ and $B$ are finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity. If $G=A*_H$ is an HNN-extension where $A$ is finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity.
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 …
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…
We prove that for a fairly large family of HNN extensions Γ, the group measure space II_1 factor L^\infty(X) \rtimes Γgiven by an arbitrary free ergodic probability measure preserving action of Γ, has a unique group measure space Cartan subalgebra up to unitary conjugacy. We deduce from this new examples of W^*-superrigid group actions, i.e. where the II_1 factor L^\infty(X) \rtimes Γentirely remembers the group action that it was constructed from.
$PD_3$-groups and HNN Extensions
研究$PD_3$-group作为HNN扩张的性质，揭示其Poincaré对偶与同调类的关系，以及$PD_3$-group在$PD_2$-group上的分裂特性。
Assume $G$ is a polycyclic group and $ϕ:G\to G$ an endomorphism. Let $G\ast_ϕ$ be the ascending HNN extension of $G$ with respect to $ϕ$; that is, $G\ast_ϕ$ is given by the presentation $$G\ast_ϕ= < G, t \ |\ t^{-1}gt = ϕ(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_ϕ}$ be the profinite completion of $G\ast_ϕ$. We prove that, for any finite discrete $\hat{G\ast_ϕ}$-module $A$, the map $H^*(\hat{G\ast_ϕ}, A)\to H^*(G\ast_ϕ,A)$ induced by the canonical map $G\ast_ϕ\to \hat{G\ast_ϕ}$ is an isomorphism.
Linear and projective boundaries of Cayley graphs were introduced in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits $g^\infty=\{g^i: i\in \mathbb N\}$, or orbits $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$, respectively, of non-torsion elements~$g$ of the group $G$, where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points $g^\infty$ and $g^{-\infty}$ in the linear boundary is bounded from below by $\sqrt{1…
HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem
本文研究了自由群的HNN扩展中字问题的多项式时间可解性，扩展到非正常子群的情况并提出了额外条件。
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