Epimorphism testing with virtually Abelian targets

几乎阿贝尔群的扭曲共轭增长级数

Twisted conjugacy growth series of virtually abelian groups

本文研究有限生成群的扭曲共轭增长级数，证明几乎阿贝尔群的该级数为可计算的$\mathbb{N}$-有理函数，并推广至相对增长级数。

Rational Growth in Virtually Abelian Groups

We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$, that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$.

Splitting the Automorphism Group of an Abelian p-Group

Let G be an abelian p-group sum of finite homocyclic groups Gi. Here, we determine in which cases the automorphism group of G splits over ker(h), where h: Aut(G)-->Xi Aut(Gi/pGi) is the natural epimorphism.

有限群满同态测试的算法

Algorithms for Finite Group Epimorphism Testing

本文针对几类结构化有限群，提出了多项式时间的群满同态测试算法，解决了这些类上满同态问题的复杂性。这些类此前已知具有多项式时间同构测试，包括具有循环补的交换正规Hall子群等。该工作推进了有限群满同态问题的算法研究。

virtually abelian 群的扭曲共轭增长

The twisted conjugacy growth of virtually abelian groups

本文研究了 virtually abelian 群上与扭曲共轭相关的多个增长函数的渐近性，并证明这些增长函数均表现出多项式渐近行为。

Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

We generalise the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $Γ$ of a right-angled Coxeter group $W_Γ$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $Out(W_Γ)$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite, or equivalently $Γ$ …

Thompson群T的无$\boldsymbol{\mathbb{Z}^2}$子群是虚自由的

$\mathbb{Z}^2$-free subgroups of Thompson's $T$ are virtually free

本文证明Thompson群T的有限生成且不含任意高阶阿贝尔群的子群为虚阿贝尔或虚自由，进而得出曲面群不能嵌入T，解答了Belk与Moore的公开问题。

虚阿贝尔群上噪声敏感性

Noise sensitivity on virtually abelian groups

研究虚阿贝尔群上非周期性随机游走的噪声敏感性，证明其在总变分下噪声敏感的充要条件是群无非零同态到无限循环群。

虚拟阿贝尔群代数集的多变量增长级数的有理性

Rationality of the multivariate growth series for algebraic sets of virtually abelian groups

本文证明了虚拟阿贝尔群代数集的多变量相对增长级数具有有理性，证实了 Evetts 和 Levine 的猜想。

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.

Commensurators of abelian subgroups and the virtually abelian dimension of mapping class groups

Let $\mathrm{Mod}(S)$ be the mapping class group of a compact connected orientable surface $S$, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup $H$ of $\mathrm{Mod}(S)$, there is a subgroup $H'$ commensurable with $H$ such that the commensurator of $H$ equals the normalizer of $H'$. As a consequence we give, for each $n \geq 2$, an upper bound for the geometric dimension of $\mathrm{Mod}(S)$ for the family of abelian subgroups of rank bounded by $n$. These results generalize work by Juan-Pineda--Truj…

Indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation group

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level~2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.

On groups and counter automata

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller-Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognised by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of cou…

CAT(0) cube complexes with flat hyperplanes

In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.

Abelian subgroups of Helly groups

We observe that abelian subgroups of Helly groups are finitely generated, and consequently, soluble subgroups of Helly groups are virtually abelian.

On group rings of virtually abelian groups

Let $Γ$ be a finitely generated torsion-free group. We show that the statement of $Γ$ being virtually abelian is equivalent to the statement that the $*$-regular closure of the group ring $\mathbb{C}[Γ]$ in the algebra of (unbounded) operators affiliated to the group von Neumann algebra is a central division algebra. More generally, for any field $k$, it is shown that $k[Γ]$ embeds into a central division algebra in case $Γ$ is virtually abelian. We take advantage of this result in order to develop a criterion for existence of units in the group ring $k[Γ]$. We develop this criterion in the pa…

非几乎阿贝尔不连续群作用与齐性空间上的真SL(2,R)作用

Non-virtually abelian discontinuous group actions vs. proper SL(2,R)-actions on homogeneous spaces

本文开发算法验证半单齐性空间上离散群作用的真性，并发现不承认真SL(2,R)作用的非几乎阿贝尔不连续群作用新例。

Free non-archimedean topological groups

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA group…

Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach

Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.

几乎阿贝尔群的有效共轭可分性

Effective conjugacy separability of virtually abelian groups

本文刻画了每个几乎阿贝尔群的共轭可分性函数，发现当扩张对应不可约表示时与剩余有限性函数相同，否则可能严格更大，并给出示例。

Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups

Given a discrete group $G$, for any integer $r\geqslant0$ we consider the family of all virtually abelian subgroups of $G$ of rank at most $r$. We give an upper bound for the Bredon cohomological dimension of $G$ for this family for a certain class of groups acting on $\mathrm{CAT}(0)$ spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of J.-F. Lafont.

近阿贝尔群中的共轭类增长

Conjugacy Class Growth in Virtually Abelian Groups

本文研究有限生成近阿贝尔群中固定共轭类在Cayley球内的计数增长，证明其渐近等价于多项式，并揭示仿射Coxeter群中增长度与反射长度的等价关系。

Non-virtually abelian anisotropic linear groups are not boundedly generated

We prove that if a linear group $Γ\subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite $S$-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are never boundedly generated. Our proof relies on Laurent's theorem from Diophantine geometry and properties of generic elements.

A virtually 2-step nilpotent group with polynomial geodesic growth

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.

Equations in virtually abelian groups: languages and growth

This paper explores the nature of the solution sets of systems of equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that the set of solutions to a system of equations forms an EDT0L language, with respect to a natural normal form. Looking at growth, we show that the growth series of the language of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group elements, we show that it has rational relative growth series with respect to any finite generating set.

Cohomological Finiteness Conditions in Bredon Cohomology

We show that any soluble group $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type $\FP_{\infty}$ must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-$\FP_n$ for some $n \leq 3$ and restrict to abelian-by-nilpotent, …

Tits alternatives for graph products

We discuss various types of Tits Alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits Alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the …

Subgroups of finite index and the just infinite property

A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is itself just infinite. If $G$ is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In particular, we show that such a group $G$ is hereditarily just infinite if and only if all maximal subgroups of finite index are just infinite. We also obtain results for certain families of virtual…

虚拟阿贝尔群中的有效方程求解、约束与增长

Effective equation solving, constraints and growth in virtually abelian groups

本文研究虚拟阿贝尔群中带组合、代数和语言约束的群方程求解，将约束转化为有理集以归约为可判定问题，并证明增长序列可有效计算。

区间交换变换群：几乎阿贝尔群的自由作用与动力学

Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups

本文研究区间交换变换群的自由与非自由作用，分类包含$\mathbb Z^2$的群的自由作用，并构造非几乎幂零的IET子群例子。

几乎阿贝尔群中的有理集：语言与增长

Rational sets in virtually abelian groups: languages and growth

本文统一并推广了关于几乎阿贝尔群中有理集增长级数有理性的结果，证明了多种重要集合类的有理性，并给出有理集的EDT0L表示。

Commensurators of abelian subgroups in CAT(0) groups

We study the structure of the commensurator of a virtually abelian subgroup $H$ in $G$, where $G$ acts properly on a $\mathrm{CAT}(0)$ space $X$. When $X$ is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of $H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$ is a $\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the action of $G$ is cellular, we show that the commensurator of $H$ is an ascending union of normalizers of finite index subgroups of $H$. We explore several special cases where the results can be strengthened and we…

On (hereditarily) just infinite profinite groups that are not virtually pro-p

A profinite group G is just infinite if every non-trivial closed normal subgroup of G is of finite index, and hereditarily just infinite if every open subgroup is just infinite. Hereditarily just infinite profinite groups need not be virtually pro-p, as shown in a recent paper of Wilson. The same paper gives a criterion on an inverse system of finite groups that is sufficient to ensure the limit is either virtually abelian or hereditarily just infinite. We give criteria of a similar nature that characterise the just infinite and hereditarily just infinite properties under the assumption that G…

Classifying spaces for the family of virtually abelian subgroups of orientable $3$-manifold groups

For a group $G$, let $\mathcal{F}_{n}$ be the family of all the subgroups of $G$ containing a subgroup isomorphic to $\mathbb{Z}^{r}$ for some $r=0,1,2,\dots ,n$ of finite index. Joecken, Lafont and Sánchez Saldaña computed the $\mathcal{F}_1$-dimension of 3-manifold groups. The goal of this article is to compute the $\mathcal{F}_n$-geometric dimension of 3-manifold groups for all $n\geq 2$.

几乎阿贝尔群中的剩余有限性增长

Residual Finiteness Growth in Virtually Abelian Groups

本文研究几乎阿贝尔群的剩余有限性增长，证明其等于 $\log^k$，其中 $k$ 由显式表达式给出，并应用于构造具有指定增长阶的群。

几乎阿贝尔群的扭曲C*-代数的核维数

Nuclear Dimension of Twisted $C^*$-Algebras of Virtually Abelian Groups

研究有限生成几乎阿贝尔群G的扭曲群C*-代数C*(G,σ)的核维数，证明其等于G的有限指数阿贝尔子群的秩，并给出Z^r上扭曲C*-代数核维数为r的充要条件。

The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.

阿贝尔-循环群的共轭比

The Conjugacy Ratio of Abelian-by-Cyclic Groups

研究了阿贝尔-循环群的共轭比，证明了该比为零除非群是虚拟阿贝尔群，并展示了非虚拟阿贝尔群的特殊情况。

Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.

一类具有少量子商的无挠可解群

Some torsion-free solvable groups with few subquotients

本研究构造出满足所有有限生成无挠亚阿贝尔子商均为虚阿贝尔的无限秩有限生成无挠可解群，证明了P. Kropholler的无限秩可解群刻画定理不存在无挠版本。

Piecewise excluding geodesic languages

The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language …

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…

Profinite groups in which centralizers are abelian

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, a rather detailed information about the finite quotient G/N is obtained.

具有自由阿贝尔顶点群的群图的直积的子群

Subgroups of the direct product of graphs of groups with free abelian vertex groups

本文将 Baumslag–Roseblade 关于自由群直积有限展示子群的结果推广到具有自由阿贝尔顶点群与循环边群的群图，并证明相关 RAAG 子群的算法问题可判定。

阿贝尔群上的代数扩张性

Algebraic expansivity on abelian groups

本文研究了阿贝尔群上端态射的代数扩张性，探讨了其基本性质、与Weiss代数熵的关系，并证明正向扩张满射仅限于有限系统，同时通过庞特里亚金对偶性展示了其与拓扑动力学的联系。

虚拟阿贝尔群的核维数与刚性结果

Nuclear Dimension and Rigidity Results for Virtually Abelian Groups

本文证明有限生成虚拟阿贝尔群的Hirsch长度等于其群C*-代数的核维数，并进一步证明晶体群性质在C*-同构下保持不变，从而推出二维晶体群是C*-超刚性的。

性质(LR)与虚自由群的嵌入定理

Property (LR) and an embedding theorem for virtually free groups

本文证明了每个虚自由群G具有Long和Reid提出的性质(LR)，即每个有限生成子群都是有限指数子群的 retract。主要方法是证明每个可数虚自由群可嵌入到有限群的双群中。

On the automorphism group of generalized Baumslag-Solitar groups

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n \times Z$)…

Algebraic characterisation of relatively hyperbolic special groups

This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $…

广义 Stallings-Bieri 群的等周行为

Isoperimetric behavior of generalized Stallings-Bieri groups

本文引入群到有限秩自由阿贝尔群满射的 $n$-split 概念，为有限展示群直积的某些共阿贝尔子群的 Dehn 函数提供界，这些子群包含并显著推广了 Stallings-Bieri 群。

所有真商群为殆幂零的群的结构

The structure of groups with all proper quotients virtually nilpotent

本文研究所有真商群均为殆幂零的群（JNN$_c$F 群），给出其有限生成 profinite 情形的刻画、结构定理及遗传性判定，并构造遗传 JNN$_c$F 群。

有限生成群中自同构轨道的计数

Counting automorphic orbits in finitely generated groups

研究有限生成群中的自同构增长函数，证明其非共轭不变量，并分类了各类群的自同构增长速率。

具有自由阿贝尔顶点群的群的图的直积的有限表现子群

Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups

本文将自由群直积有限表现子群的相关结构定理推广到二维凝聚右角阿廷群（RAAGs）直积，刻画了其有限表现及$FP_n$型子群的结构，并证明这类子群的多重共轭问题与成员问题可判定。

Conjugacy languages in groups

We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.