# Questions tagged [nilpotent-groups]

The nilpotent-groups tag has no usage guidance.

79
questions

0
votes

1
answer

117
views

### 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 ...

6
votes

1
answer

203
views

### 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 ...

2
votes

0
answers

65
views

### 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 $\gamma_{k+1}(F_n)$ ...

4
votes

0
answers

179
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 ...

5
votes

0
answers

253
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|}\...

3
votes

1
answer

108
views

### 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
votes

0
answers

125
views

### 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 ...

0
votes

0
answers

154
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, ...

0
votes

1
answer

82
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/...

1
vote

0
answers

54
views

### 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 $\...

2
votes

1
answer

132
views

### 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}(...

3
votes

0
answers

56
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\} \...

3
votes

0
answers

89
views

### 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\...

5
votes

0
answers

144
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 ...

9
votes

2
answers

219
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 $x \in M$, ...

5
votes

1
answer

198
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 $< p$,...

7
votes

1
answer

163
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 ...

6
votes

1
answer

208
views

### 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 ...

7
votes

1
answer

231
views

### 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 $\...

6
votes

1
answer

218
views

### 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 $...

5
votes

1
answer

158
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 $...

0
votes

0
answers

67
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 ...

3
votes

1
answer

123
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 ...

1
vote

1
answer

86
views

### Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...

1
vote

1
answer

173
views

### 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^{}}}) ...

4
votes

0
answers

129
views

### 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 $\...

6
votes

1
answer

550
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/...

1
vote

0
answers

112
views

### 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
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...

3
votes

1
answer

225
views

### 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
...

1
vote

0
answers

46
views

### 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 ...

5
votes

0
answers

166
views

### 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 $\geq 3$...

9
votes

1
answer

562
views

### 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 $N$ ...

5
votes

2
answers

862
views

### 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 ...

2
votes

0
answers

123
views

### Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form
$$\beta_y(x)=d(x,y)-d(w,y).$$
The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...

11
votes

0
answers

244
views

### Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...

2
votes

1
answer

74
views

### Defect of subnormality and repeated normalizer series

Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U_i)_{1\le i\le n}$ such that $U_1=S$, $U_i$ normal in $U_{i+1}$ and $U_n=G$ is called the ...

2
votes

0
answers

65
views

### The nilpotentizer of the Hirsch-Plotkin radical in a finitely generated poly-(locally nilpotent) group

Let $G$ be a radical group (a group having a normal series with locally nilpotent factors) and $H$ its Hirsch Plotkin racial (i.e the locally nilpotent radical of $G$). It is well known that $C_G(H)\...

9
votes

0
answers

418
views

### (Torsion in) homology of free nilpotent groups

It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...

5
votes

1
answer

228
views

### Harmonic analysis on nilpotent Lie groups and the Campbell-Hausdorff formula

I am trying to understand the non-commutative analysis for nilpotent Lie groups, so I've been reading Corwin's and Greenleaf's book on the representation theory of nilpotent groups and going through ...

14
votes

3
answers

1k
views

### Explicit formulas for Carnot-Carathéodory distances on Carnot groups

Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = ...

25
votes

0
answers

941
views

### Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real ...

1
vote

1
answer

346
views

### Compact non-connected nilpotent Lie subgroup of $O(n)$?

Let $G$ be a compact non-connected nilpotent Lie subgroup of $O(n)$. We know that $G_0$, its identity component, is always a torus. Is it true that $G_0$ is always central in $G$?
What about general $...

1
vote

0
answers

141
views

### Reference request for the list of nilpotent subgroups of SU(2)?

It's not hard to show that all non-abelian nilpotent subgroups of $SU(2)$ are actually finite and in fact are conjugate to one of the generalized quaternion groups of order a power of two, $$Q_{2^n} =...

2
votes

0
answers

71
views

### Operators associated with unitary representations of nilpotent Lie group

Let $G$ be a nilpotent Lie Group, and $\pi:G\to B(\mathcal H)$ be an irreducible unitary representation on the Hilbert space $\mathcal H$. One can use the Bochner integral to define a linear map as ...

1
vote

1
answer

97
views

### Relation between flat and nilpotent structures on fibers?

When collapsing Riemannian manifolds under suitable curvature conditions, two types of structure arise on the fibers: flat structures and nilpotent structures. This depends on the scale at which one ...

4
votes

0
answers

130
views

### Order problem in nilpotent groups

Let $G$ be a f.g. nilpotent group. I wanted to know if the order problem (given $g \in G$, deciding if there exists $n$ s.t. $g^n=e$) is decidable in $G$? In such a group, the word problem is ...

7
votes

0
answers

415
views

### Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure

Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...

2
votes

2
answers

328
views

### Nilpotency of Lie Algebra from Structure Constants

Suppose we have a Lie algebra with structure constants
$$\mathrm{d}e^i=\sum_{j<k}a_{ijk}e^j\wedge e^k$$
for some coefficients $a_{ijk}$.
In this setting, how may be checked (perhaps ...

11
votes

2
answers

704
views

### Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...

12
votes

1
answer

2k
views

### Some questions about the Malcev completion

Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...