Virtually abelian quotients of random groups

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 $G$.

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 $Γ$ …

Coxeter quotients of the automorphism group of a Coxeter group

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Γ$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_Γ)$ virtually surjects onto some infinite Coxeter group. In particular, the group ${\rm Aut}(W_Γ)$ is virtually indicable and therefore does not have Kazhdan's property (T).

The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n/[P_n,P_n]$

Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.

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。

Virtually special non-finitely presented groups via linear characters

We present a new method for showing that groups are virtually special. This is done by considering finite quotients and linear characters. We use this to show that an infinite family of groups, related to Bestvina-Brady groups and branching, provides new examples of virtually special groups outside of a hyperbolic context.

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。

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。

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…

具有阿贝尔缺陷群和素数阶惯性商的2块

2-blocks with abelian defect groups and inertial quotient of prime order

本文分类了所有缺陷群为阿贝尔且惯性商为素数阶的2块，证明了布鲁埃关于阿贝尔缺陷群猜想在所考虑的块中成立。

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.

Abelian and metabelian quotients of surface braid groups

In this paper we study abelian and metabelian quotients of braid groups on oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell-Neuwirth fibrations.

Fitting quotients of finitely presented abelian-by-nilpotent groups

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

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。

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.

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。

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。

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…

由半扭转立方生成的群的阿贝尔商

Abelian quotients for groups generated by cubed half-twists

本文研究由半扭转立方生成的辫群子群，构造其非循环无挠阿贝尔商，并计算3股和4股情形下的阿贝尔化及3级同余子群的阿贝尔化。

Approximating a group by its solvable quotients

The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a group has solvable Farb growth which is at most polynomial in $\log(n)$ if and only if the group is polycyclic and virtually nilpotent. We also give new results concerning approximating oriented surface groups by nilpotent quotients. As a consequence of this, we prove that a natural number $C$ exists so that any nontrivial element of the $Ck$th term of the lo…

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.

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.

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.

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.

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。

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。

On subgroup conjugacy separability in the class of virtually free groups

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.

具有虚拟阿贝尔靶标的满同态测试的复杂性

On the complexity of epimorphism testing with virtually abelian targets

该研究证明了有限表现群到几类虚拟阿贝尔群及特定有限群的满同态测试问题的$\textsf{NP}$完全性，扩展了相关可判定性的靶标群范围，补充了已有研究结论。

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.

具有阿贝尔商的直角阿廷群的有限生成核

Finitely presented kernels of right-angled Artin groups with abelian quotients

研究具有阿贝尔商的直角阿廷群的核何时有限生成及有限表示，给出明确生成集与表示，还推导出其特征标核的有限表示，推广了相关群的表示。

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, …

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。

Virtually Fibering Right-Angled Coxeter Groups

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb H^4$ with fundamental domain the $120$-cell or the $24$-cell.

Almost Abelian Lie groups, subgroups and quotients

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified a…

Abelian quotients and orbit sizes of finite groups

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at least $|G/G'|$ on $V$. This generalizes earlier work of the authors, where the same bound was proved under the additional hypothesis that $G$ is solvable. For completely reducible modules it also strengthens the 1989 result $|G/G'|<|V|$ by Aschbacher and Guralnick.

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…

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.

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。

On the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$

We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb{Z}/2\mathbb{Z}) = \mathrm{L}_n(2)$, thus confirming a conjecture of Mecchia--Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut(F_n)}$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of…

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…

$PL^+[0,1]$ 的有限生成子群的恰无限商

Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$

本文证明 Richard Thompson 群 F 及两类分段同胚群的有限生成子群的恰无限商实际上交换，从而回答 Grigorchuk 的问题，并给出众多非顺从群例子。

关于在虚拟幂零商上的纤维积的有限性性质

Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients

本文针对虚拟幂零商群，证明了纤维积有限性定理的对称加强版，给出了类型F_{min{k+l+1,m,n}}的精确结果，并提供了Virtual Surjections Theorem的替代证明。

Profinite completions of Burnside-type quotients of surface groups

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.

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…

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H_i} of subgroups of G with trivial intersection, such that the union generates G and all the quotients G/H_i have the same e…

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。

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。

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。