Configuration of nilpotent groups and isomorphism

The problem of the classification of the nilpotent class 2 torsion free groups up to the geometrically equivalence

In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s) groups is Noethe…

Isomorphisms of Cayley graphs on nilpotent groups

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. F…

Nilpotent centralizers and Springer isomorphisms

Let G be a semisimple algebraic group over a field K whose characteristic is very good for G, and let sigma be any G-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map sigma is known as a Springer isomorphism. Let y in G(K), let Y in Lie(G)(K), and write C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y are smooth group schemes over K. The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the …

Regular dessins uniquely determined by a nilpotent automorphism group

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…

Compatible filters with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

半刚性与指数为三的 nilpotent 半群的计数

Semirigidity and the enumeration of nilpotent semigroups of index three

本文研究了指数为三的 nilpotent 半群的计数问题，提出半刚性概念并给出了上界公式，改进了同构类数量的下界。

Relatively free nilpotent torsion-free groups and their Lie algebras

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…

Power graphs of all nilpotent groups

The directed power graph $\vec{\mathcal G}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph $\mathcal G(\mathbf G)$ of the group $\mathbf G$ is the underlying simple graph. In this paper, we prove that Prüfer group is the only nilpotent group whose power graph does not determine the directed power graph up to isomorphism. Also, we present a group $\mathbf G$ with quasicyclic torsion subgroup that is determined by its power graph up to isomorphism, i.e. such that $\mathcal G(\mathbf H)\cong\mathcal …

Levi decomposition of nilpotent centralisers in classical groups

We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].

Isomorphism invariants of restricted enveloping algebras

Let $L$ and $H$ be finite-dimensional restricted Lie algebras over a perfect field $\F$ such that $u(L)\cong u(H)$, where $u(L)$ is the restricted enveloping algebra of $L$. We prove that if $L$ is $p$-nilpotent and abelian, then $L\cong H$. We deduce that if $L$ is abelian and $\F$ is algebraically closed, then $L\cong H$. We use these results to prove the main result of this paper stating that if $L$ is $p$-nilpotent, then $L/L'^p+γ_3(L)\cong H/H'^p+γ_3(H)$.

Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups

Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$ encoding, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. We show that the domains of convergence and meromorphic continuation of these zeta functions of groups $\mathbf{G}(\mathcal{O})$ are independent of the number fi…

关于到自由群的同构及相关问题

On isomorphisms to a free group and beyond

本文针对群论中难解的无限有限表现群同构问题，研究了有限表现群与自由群等特定群的同构判定，以及有限表现群嵌入给定极限群的算法问题。

Symbolic computation of Schur multipliers with an application to the groups of order dividing $p^6$

We describe an algorithm to compute the Schur multipliers of all nilpotent Lie $p$-rings in the family defined by a symbolic nilpotent Lie $p$-ring. Symbolic nilpotent Lie $p$-rings can be used to describe the isomorphism types of $p$-groups of order $p^n$ for $n \leq 7$ and all primes $p \geq n$. We apply our algorithm to compute the Schur multipliers of all $p$-groups of order dividing $p^6$.

Two-sided configuration equivalence and isomorphism

The concept of configuration was first introduced to give a characterization for the amenability of groups. Then the concept of two-sided configuration was suggested to provide normality to study the group structures more efficiently. It has been interesting that for which groups, two-sided configuration equivalence would imply isomorphism. We introduce a class of groups, containing polycyclic and FC groups, which for them, the notions of two-sided configuration equivalence and isomorphism coincide.

具有同构有限商的小 Hirsch 长度无挠幂零群

Torsion-free nilpotent groups of small Hirsch length with isomorphic finite quotients

本文刻画了 Hirsch 长度不超过 5 的有限生成无挠幂零群中具有相同有限商同构类的群集合，并证明当 Hirsch 长度至少为 4 时，这类集合的基数可任意大。

Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.

Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.

通过轨道计数枚举 nilpotent 环

Enumeration of Nilpotent Loops by Orbit Counting

通过引入仿射自同构群，研究 nilpotent 环的中央扩张并高效计数其轨道。

广义Baumslag-Solitar群的同构问题：不变量与灵活构型

On the isomorphism problem for generalized Baumslag-Solitar groups: invariants and flexible configurations

本研究针对具有一个拟共轭类且完全支撑带隙的广义Baumslag-Solitar（GBS）群，引入一族不变量，证明其同构问题可判定，且该不变量可完全刻画此类群的同构性。

Configuration Equivalence is not Equivalent to Isomorphism

Giving a condition for the the amenability of groups, Rosenblatt and Willis, first introduced the concept of configuration. From the beginning of the theory, the question whether the concept of configuration equivalence coincides with the concept of group isomorphism was posed. We negatively answer this open question by introducing two non-isomorphic, solvable and hence amenable groups which are configuration equivalent. Also, we will study some types of subgroups in configuration equivalent groups. In particular, we will prove this conjecture, due to Rosenblatt and Willis, that configuration …

Quadratic envelope of the category of class two nilpotent groups

We exhibit an extension of the category of class two nilpotent groups. It has the same objects but, unlike the latter, its morphisms are closed under pointwise addition of maps. At the same time the class of its morphisms is much smaller than the class of all maps. In fact, the morphisms are quadratic maps of very special kind. It is indicated that classification of various class two nilpotent groups up to isomorphism in this larger category can give interesting results.

The Probability Distribution of Word Maps on Finite Groups

Word maps provide a wealth of information about finite groups. We examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group. We show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size. Moreover, we show that probability distributions themselves are sufficient to identify nilpotent groups, and these same distributions can be used to determine abelian groups up to isomorphism. In addition we answer a question of Amit and Vishne.

Embeddings into pro-p groups and the construction of parafree groups

A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two constructions, namely amalgamated products with cyclic amalgams and cyclic HNN extensions.

幂零群、可解群与可因子化逆半群

Nilpotent groups, solvable groups and factorizable inverse monoids

本文引入逆半群中的次中心与中心幂等元序列，证明了可因子化逆幺半群中此类序列的同构性（类比Jordan-Hölder定理），并据此定义了G-幂零与G-可解逆幺半群，为研究幂零群和可解群提供了新途径。

The Congruence Subgroup Problem for finitely generated Nilpotent Groups

The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…

Transporting cohomology in Lazard correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent $\mathbb{Z}_p$-Lie algebras of nilpotency class smaller than $p$. Denote by $H_{Gr}^i$ and $H_{Lie}^i$ the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for $i=0$, $1$ and $1$, and for a given category of modules the cohomology functors $H_{Gr}^i\circ \textbf{exp}$ and $H^i_{Lie}$ are naturally equivalent. A similar result is proven for $i=3$ and the r…

Isomorphisms between centers of integral group rings

For finite nilpotent groups $G$ and $G^{\prime}$, and a $G$-adapted ring $S$ (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings $SG$ and $SG^{\prime}$ is monomial, i.e., maps class sums in $SG$ to class sums in $SG^{\prime}$ up to multiplication with roots of unity. As a consequence, $G$ and $G^{\prime}$ have identical character tables if and only if the centers of their integral group rings $\mathbb{Z} G$ and $\mathbb{Z} G^{\prime}$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

Solubility of groups can be characterized by configuration

The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. We show that group properties of being soluble or FC can be characterized by configuration sets. Then we investigate some condition on configuration pairs, which leads to isomorphism. We introduce a somewhat different notion of configuration equivalence, namely strong configuration equivalence, and prove that strong configuration equivalence coincides with isomorphism.

幂图的同构问题与 Cameron 的一个问题

The Isomorphism Problem of Power Graphs and a Question of Cameron

本文研究有限幂零群产生的幂图、有向幂图和增强幂图的同构问题，给出无需底层群的多项式时间算法，并解决了 Cameron 关于从幂图或增强幂图恢复有向幂图的问题。

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…

On Borel complexity of the isomorphism problems for graph-related classes of Lie algebras and finite p-groups

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarit…

具有阿贝尔Sylow子群的群的多项式时间同构测试

Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups

本文针对具有阿贝尔Sylow子群的群（A-群）提出多项式时间同构测试算法，推广了Babai和Qiao关于可解群的结果，并利用了其计算方法和A-群的结构性质。

Isometries of nilpotent metric groups

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equipped with Carnot-Carathéodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent th…

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 $F_n \times Z$)…

Localization, metabelian groups, and the isomorphism problem

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and othe…

A Lie Algebra Correspondence for a Family of Finite p-Groups

For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent p^n-element Lie algebras L over the truncated polynomial ring F_p[T]/(T^n) in which T[L,L]=0.

Homology, lower central series, and hyperplane arrangements

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.

遍历映射与幂零李群的上同调

Ergodic maps and the cohomology of nilpotent Lie groups

研究幂零群上同调如何受李普希茨映射影响，给定单连通幂零李群间光滑李普希茨映射\(f\)，存在遍历映射\(\psi\)，其拉回有顺从平均且是同调代数同态，\(f\)为准等距时，\(\psi\)也是且\(\overline{\psi^*}\)是同构，推广简化相关定理证明。

关于广义Baumslag-Solitar群的同构问题：角度

On the isomorphism problem for generalized Baumslag-Solitar groups: angles

本文为广义Baumslag-Solitar群引入新的同构不变量“极限角”，具有动力学解释，并用于分类一个顶点两条边的GBS群。

The power graph of a torsion-free group

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if $y$ is a power of $x$. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, …

钻石：同调与群的中央列

Diamonds: Homology and the Central Series of Groups

研究通过群同调条件，证明非同构数域和超曲面的基本群可有同构的universal nilpotent quotients，揭示了伽罗瓦作用与nilpotent表示论的关系。

幂零群拟等距的上同调与乘法间隙

Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups

该研究在七维无挠幂零格上构造两个等距渐近锥的字度量，利用拟等距的凸不变量与上同调相容性，证明其拟等距映射存在至少1+ε的乘法间隙。

On the Weisfeiler-Leman Dimension of Finite Groups

In comparison to graphs, combinatorial methods for the isomorphism problem of finite groups are less developed than algebraic ones. To be able to investigate the descriptive complexity of finite groups and the group isomorphism problem, we define the Weisfeiler-Leman algorithm for groups. In fact we define three versions of the algorithm. In contrast to graphs, where the three analogous versions readily agree, for groups the situation is more intricate. For groups, we show that their expressive power is linearly related. We also give descriptions in terms of counting logics and bijective pebbl…

Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell's unitary nilpotent groups UNil_*(Z[F];Z[F],Z[F]) have an induced isomorphism to the quotient of UNil_*(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial group. The broader scope is the study of the L-theory of virtually cyclic groups, based on the Farrell--Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2…

关于群上定义的图的问题的复杂性

On the Complexity of Problems on Graphs Defined on Groups

本文研究群上定义的图（如幂图）上若干图问题的复杂性，证明在ETH下同构不变问题非NP完全，并给出加权最大割的NP完全性及图基序的难解性，同时探讨幂图识别的多项式算法。