J. O. Button
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.
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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。
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。
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…
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。
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。
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, …
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 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.
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 …
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.
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…
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…
The Lodha-Moore groups provide the first known examples of type F_\infty groups that are non-amenable and contain no non-abelian free subgroups. These groups are related to Thompson's group F in certain ways, for instance they contain it as a subgroup in a natural way. We exhibit decompositions of four Lodha-Moore groups, G, G_y, {_y}G and {_y}G_y, into ascending HNN extensions of isomorphic copies of each other, both in ways reminiscent to such decompositions for F and also in quite different ways. This allows us to prove two new topological results about the Lodha-Moore groups. First, we pro…
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.
The Lamplighter Group is Not Semistable at Infinity
该研究解决了四十余年的公开问题，首次证实有限生成的Lamplighter群在无穷远处不半稳定，并指出其升HNN扩张Extended Lamplighter群虽为有限表现，但经Silkin结果证明在无穷远处单连通，为寻找有限表现非半稳定群提供了研究方向。
Non-linear, solvable, residually $p$ groups
本文构造了有限生成非线性可解剩余2群的第一个例子，填补了从剩余有限群到剩余p群这一研究方向的空白。
Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result …
We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts: 1) We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many…
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.
Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\Emb(S)$ has finitely many isomorphism classes. For instance, we give an example where $\Emb(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\Emb(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.
Let G be an infinite cyclic extension, 1 -> B -> G -> Z -> 1, of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain ascending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|> 1. We construct a model for Evc(G), the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimum dimension of Evc(G). We construct a 2-dimensional model for Evc(G) where G is a soluble Baumslag-Solita…
Groups acting on horocyclic products
研究群在horocyclic积上的几何作用，发现其结构取决于该积空间视觉边界的连通性，可能为有限生成虚-nilpotent群的上升HNN扩张或非有限呈现群。
Liftable self-similar groups and scale groups
本文通过将局部域的等距与伸缩群等同于正则树自同构群的子群，引入可提升自相似群并证明其升HNN扩张的闭包构成尺度群，同时给出多个嵌入2-进数域伸缩群的具体例子。
A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if |det(M)|=1. We give a complete classification of the nonpolycyclic groups Gamma_M up to quasi-isometry: given n x n integer matrices M,N with |det(M)|, |det(N)| > 1, the groups Gamma_M, Gamma_N are quasi-isometric if and only if there exist positive integers r,s such that M^r, N^s have the same absolute Jordan form. We also prove quasi-isometric rigidity: if Gamma…
Given a Kaehler group $G$ and a primitive class $ϕ\in H^1(G;Z)$, we show that the rank gradient of $(G;ϕ)$ is zero if and only if Ker $ϕ$ is finitely generated. Using this approach, we give a quick proof of the fact (originally due to Napier and Ramachandran) that Kaehler groups are not properly ascending or descending HNN extensions. Further investigation of the properties of Bieri-Neumann-Strebel invariants of Kaehler groups allows us to show that a large class of groups of orientation-preserving PL homeomorphisms of an interval, which generalize Thompson's group $F$, are not Kaehler.
Non-MF groups and non-finite full group C*-algebras
该研究针对具有性质(T)且存在单射非满自同态的群Γ及其升HNN扩张G，构造W=(⊕_{G/Γ}ℤ/2ℤ)⋊G，证明W非MF群且C*(G)非有限C*-代数，论证由ChatGPT 5.6 Sol生成并完善。
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