Questions tagged [nilpotent-groups]
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96 questions
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Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\mathbb{Z}/n\mathbb{Z}[x_1,...,x_k]/\langle ...
- 25.7k
1
vote
0
answers
118
views
Criterion for when certain closed 2-forms generates all n-forms
Let $\omega_1,\dots,\omega_r$ be closed 2-forms in $\Bbb R^m$.
I want to find a smallest number $n\ge1$ such that every $(n+1)$-form $\eta$ can be written as $\eta=\beta_1\wedge\omega_1+\dots+\beta_r\...
- 1,511
5
votes
1
answer
359
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Why do we care about residually solvable/nilpotent groups?
I'm interested in a certain property $X$. In the introduction to basically every paper on $X$ there's a paragraph that goes something like:
$X$ is related to residually-solvable groups in this way, ...
- 491
7
votes
1
answer
542
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Khovanskii's theorem in nilpotent groups?
In $\mathbb{Z}^d$, a classical result of Khovanskii states that for any finite set $A \subseteq \mathbb{Z}^d$, the sumset $hA := A + \cdots + A$ eventually agrees with a polynomial in $h$; that is, $|...
- 962
4
votes
1
answer
272
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Outer automorphism groups of nilpotent groups
I am interested in the class of finitely generated nilpotent groups. It seems that many things are known about the automorphism groups of such groups. However, I could not find much information on ...
- 141
8
votes
1
answer
334
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Finite index subgroup of nilpotent group formed by squares
Let $G$ be a finitely generated nilpotent group. Is there a finite index subgroup $H$ of $G$ such that $H\subseteq\{g^2;g\in G\}$?
Context: I considered this question while studying `actions' of the ...
- 13.1k
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0
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86
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Positivity of polynomial ergodic averages in nilpotent groups
Let $(X,\mu)$ be a probability space with an action $(T_g)_{g\in G}$ of a group $G$ by unitary transformations. Theorem 1.1 in Zorin-Kranich's paper implies that, if $G$ is nilpotent, $H$ is a (...
- 13.1k
0
votes
1
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146
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Normalizer of $\varphi$-invariant subgroup is $\varphi$-invariant
In a paper I'm writing, I'm proving the following two results:
Let $N$ be a finitely generated torsion-free nilpotent group and $H$ a subgroup of $N$. Suppose $\varphi$ is an automorphism of $N$ such ...
- 129
2
votes
0
answers
77
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Homomorphism to a finite p-group/Lie ring Q: estimate on |Q|
Let $L$ be a finitely generated, torsion-free nilpotent group. Furthermore, assume it is also a Lie ring, i.e. a Lie algebra but over $\mathbb{Z}$. The correspondance between $L$ as a group and $L$ as ...
- 61
11
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1
answer
472
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A question on groups having a subgroup which fixes a vector in every irreducible representations
Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
- 419
3
votes
0
answers
132
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Lattice in a simply connected nilpotent Lie group
Given a connected and simply connected nilpotent Lie group $N$ with a left invariant metric, we assume that there is a lattice $\Gamma$ of $G$. Let $B_1(e)$ be the $1$-ball at the identity element in $...
- 31
3
votes
1
answer
534
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 967
9
votes
3
answers
1k
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Finding a nilpotent, infinite, f.g., virtually abelian, irreducible integer matrix group
I was wondering if anyone here knows of an example of a group $M \leq \mathrm{GL}_n(\mathbb{Z})$ which is
nilpotent,
infinite,
finitely generated,
virtually abelian,
irreducible (over $\mathbb{Z}$ or ...
- 5,792
6
votes
0
answers
213
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Equation in a nilpotent group
Let $G$ be a nilpotent group of class at most $r$
(that is, $\gamma^{r+1}G=1$).
Let elements $g_1,\dotsc,g_n\in G$ be fixed.
We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
- 809
7
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0
answers
292
views
Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
- 26.3k
2
votes
0
answers
169
views
Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
2
votes
1
answer
280
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Commensurability classes of subgroups of a nilpotent group
Here is a question I stumbled upon in my research.
Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?
Recall that two ...
- 3,403
10
votes
1
answer
593
views
Can automorphism equivalence in a free group be detected in a nilpotent quotient?
If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$.
Let $F = F_2$ be the free group on two ...
- 11.5k
0
votes
1
answer
277
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What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
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1
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310
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Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
- 243
2
votes
0
answers
121
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Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
4
votes
0
answers
262
views
A different approach to proving a property of finite solvable groups
Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution!
I asked this on math.stackexchange a couple of days ago, but it didn't attract any ...
- 2,393
5
votes
0
answers
384
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
- 27.6k
3
votes
1
answer
160
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Bounding size of group by number of generators, order of elements, and nilpotency class (Restricted Burnside's)
I have been looking at some papers on the "restricted Burnside problem". On page 4 of Vaughan-Lee and Zelmanov's survey, "Bounds in the restricted Burnside problem", I think they ...
- 3,509
3
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0
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249
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Semi-direct products and associated graded Lie algebras
Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...
- 967
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votes
0
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218
views
When does this commutative non-associative algebra have nilpotent elements?
Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients
$(a_0, \dotsc, ...
- 799
0
votes
1
answer
187
views
Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 11
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0
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105
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Is center of connected nilpotent Lie group lattice hereditary?
This might be a stupid question, but I couldn't find a reference/explanation.
Let $G$ be a connected nilpotent Lie group, and $\Gamma$ a lattice in it.
If $Z$ is the center of $G$, is it true that $\...
- 143
2
votes
1
answer
169
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Element that is in $\phi^{-1}(Z(F (G/F(G)))$
I'm studying an article but I'm not able to understand one of his statements. I have the following hypotheses:
$G$ is a solvable group with trivial center, $J=\phi^{-1}(F(G/F(G)))$ and $J_2=\phi^{-1}(...
- 123
4
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0
answers
91
views
Dilation of a Voronoi cell in a nilpotent Lie group
Let $N$ be a nilpotent Lie group equipped with some left-invariant metric $d$ and a family of dilations $\{\delta_t\}_{t \in \mathbb{R}_+}$. Suppose that there is a collection of points $\{x_i\} \...
- 566
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119
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About the nilpotency of a subgroup
Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...
- 2,148
5
votes
0
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216
views
Finitely generated nilpotent groups with hyperbolic automorphisms
$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$
Let $G$ be a finitely generated nilpotent group.
We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...
- 8,619
9
votes
2
answers
320
views
Residual finiteness for modules over group rings
Let $G$ be a finitely generated residually finite group and let $M$ be a finitely generated $\mathbb{Z}[G]$-module.
Question: Must $M$ be residually finite in the sense that for all nonzero $x \in M$, ...
- 93
5
votes
1
answer
339
views
Local vs global nilpotence class (Lazard correspondence)
The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$,...
- 3,790
7
votes
1
answer
278
views
Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group
Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the ...
- 153
6
votes
1
answer
465
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When is a nilpotent Lie algebra isomorphic to the associated graded of its lower central series?
All Lie algebras in this question are finite-dimensional and defined over a field $k$ of characteristic $0$ which I'm happy to take to be $\mathbb{R}$ or $\mathbb{C}$.
$\DeclareMathOperator\gr{gr}$Let ...
- 437
7
votes
1
answer
378
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Lower central series vs torsion-free lower central series
$\newcommand\tf{\text{tf}}\newcommand\tor{\text{tor}}$Let $G$ be a finitely generated group. Let $\gamma_k(G)$ denote the $k$th term in the lower central series for $G$, so $\gamma_1(G) = G$ and $\...
- 437
6
votes
1
answer
323
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Betti numbers and lower central series quotients of finite-index subgroups of nilpotent groups
Let $G$ be a finitely generated nilpotent group and let $A\le G$ be a finite-index subgroup. I have two questions about $A$:
Is it true that the inclusion map $A \rightarrow G$ induces isomorphisms $...
- 437
5
votes
1
answer
261
views
Example of an invariant metric on a nilpotent group which is not asymptotically geodesic
Let $X$ be a metric space. We say that $X$ is asymptotically geodesic if for all $\epsilon > 0$, there exists $R > 0$ such that, for all $x,y \in X$, there exists some finite sequence of points $...
- 305
0
votes
0
answers
100
views
Estimate of the nilpotency class from the subgroup
Let $G$ be a nilpotent group and $H \vartriangleleft G$ a normal subgroup such that $[G:H] \le m$.
Assume $H$ has the nilpotency class $ \le n$. Can we show the nilpotency class of $G$ is bounded by a ...
- 2,585
4
votes
1
answer
235
views
Centre of solvable locally nilpotent groups
This question is motivated by two examples of locally nilpotent groups which I came across (see below).
Question: Given an infinite solvable and locally nilpotent group $G$, does $G$ have an infinite ...
- 4,726
1
vote
1
answer
249
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Presentations of groups of order $p^4$
In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
- 391
4
votes
0
answers
147
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Linear vs algebraic unipotent quotient stacks
Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either
Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations
Type 2: $G$ acts on $\...
- 3,822
6
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1
answer
814
views
Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?
I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
user121195
1
vote
0
answers
169
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Nilpotency of topological groups
A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...
- 991
3
votes
1
answer
319
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Free Lie algebra and nilpotent groups in Rothschild and Stein's paper
In
Rothschild, Linda Preiss; Stein, Elias M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(1976), 247-320 (1977). ZBL0346.35030. PDF at archive.ymsc.tsinghua.edu.cn
they ...
- 573
1
vote
0
answers
69
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Descending FC series
In analogy to the central series one can define a FC series as a sequence $A_i$ of normal subgroups such that
$$
\{1\} = A_0 \lhd A_1 \lhd A_2 \lhd \cdots \lhd A_n = G
$$
such that $A_{i+1}/ A_i$ is ...
- 4,726
5
votes
0
answers
218
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Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
- 2,190
10
votes
1
answer
875
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What is the simplest known finite presentation of a free nilpotent group?
Finitely generated nilpotent groups are always finitely presented. This is true for abelian groups, and can be shown by induction for nilpotent ones, using the classical lift of a presentation of $N$ ...
- 273
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votes
2
answers
1k
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Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
- 51