On the linearity of HNN-extensions with abelian base group

HNN decompositions of the Lodha-Moore groups, and topological applications

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…

On Linear Representations of Some Extensions

Yu. I. Merzljakov developed a method of splittable coordinates which helps to verify the linearity of some groups, he established some fundamental results using this method. In this paper we use the method of splittable coordinates and find some sufficient condition under which the semi--direct product of two linear groups is linear. As consequence we get linearity of some HNN-extensions of a free group, linearity of the holomorph of the braid group B_n, n >1, and linearity of some Artin groups. In all cases we construct faithful linear representations in the explicit form.

自由阿贝尔乘自由群及相关族的轨道问题

Orbit problems for free-abelian times free groups and related families

该研究证明自由阿贝尔乘自由（FATF）群Fn×Z^m的自同态的Brinkmann问题、扭曲共轭问题可判定，且单同态的双侧Brinkmann共轭问题可判定，进而推导出自FATF群的升HNN扩张的共轭问题可判定。

An ascending HNN extension of a free group inside SL(2,C)

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…

Linearity problem for non-abelian tensor products

In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products $G \otimes H$ and tensor squares $G \otimes G$. Using these results we prove that tensor squares of some groups with one relation and some knot groups are linear. We prove that the Peiffer square of finitely generated linear groups is linear. At the end we construct faithful linear representations for the non-abelian tensor square of free group and free nilpotent group.

Automaticity for graphs of groups

In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the associated automatic subgroups, subject to further geometric conditions. These results are proved in the general context of fundamental groups of graphs of groups. The hypotheses of our closure results are satisfied in a variety of examples such as Artin groups of sufficiently large type, Coxeter groups, virtually abelian groups, and groups that are hyperbolic relati…

Deficiency and abelianized deficiency of some virtually free groups

Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $Γ_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $Γ_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $Γ_{m,n}$ has a relation gap.

A Cubical Flat Torus Theorem and the Bounded Packing Property

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.

Group rings of countable non-abelian locally free groups are primitive

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the known result for the countable case to the general cardinality case. In order to prove the main theorem, we state some graph-theoretic results and apply them to to Formanek's method.

The dynamics and geometry of free group endomorphisms

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…

Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups

We prove that if a right-angled Artin group $A_Γ$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_Γ$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_Γ$ and $A_Δ$ are commensurable and $Γ$ has no separating $k$-cliques for any $k\le n$ then neither does $Δ$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the…

On the asymptotic geometry of abelian-by-cyclic groups

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…

pro-$\mathcal{C}$群的循环分裂

Cyclic splittings of pro-$\mathcal{C}$ groups

本文将Rips-Sela关于沿循环子群作合冲自由积或HNN扩张分裂的定理推广到满足封闭条件的pro-$\mathcal{C}$群。

Groups acting freely on $Λ$-trees

A group is called $Λ$-free if it has a free Lyndon length function in an ordered abelian group $Λ$, which is equivalent to having a free isometric action on a $Λ$-tree. A group has a regular free length function in $Λ$ if and only if it has a free isometric action on a $Λ$-tree so that all branch points belong to the orbit of the base point. In this paper we prove that every finitely presented $Λ$-free group $G$ can be embedded into a finitely presented group with a regular free length function in $Λ$ so that the length function on $G$ is preserved by the embedding. Next, we prove that every f…

有限 Morley 秩的交换群上的有限群作用

Finite group actions on abelian groups of finite Morley rank

本文在有限 Morley 秩范畴内研究有限群对无限交换群的作用，为移除“尖锐性”假设及证明单代数群作用的线性猜想提供关键工具。

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product $G = H *_A K $ with $[H:A] \geq [K:A] \geq 2$, the Schreier graph with respect to $H$ or $K$ turns out to be non-amenable if and only if $[H:A] \geq 3$. Moreove…

Non-split linear sharply $2$-transitive groups

We give examples of countable linear groups in $SL_{n}(R)$ for $n \ge 3$, with no nontrivial normal abelian subgroups, that admit a faithful sharply 2-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.

Classifying Spaces with Virtually Cyclic Stabilisers for Certain Infinite Cyclic Extensions

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…

Group extensions over infinite words

We construct an extension $E(A,G)$ of a given group $G$ by infinite non-Archimedean words over an discretely ordered abelian group like $Z^n$. This yields an effective and uniform method to study various groups that "behave like $G$". We show that the Word Problem for f.g. subgroups in the extension is decidable if and only if and only if the Cyclic Membership Problem in $G$ is decidable. The present paper embeds the partial monoid of infinite words as defined by Myasnikov, Remeslennikov, and Serbin (Contemp. Math., Amer. Math. Soc., 378:37-77, 2005) into $E(A,G)$. Moreover, we define the exte…