Questions tagged [nilpotent-groups]

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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 $\...
2 votes
1 answer
128 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
45 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
85 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
123 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 ...
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9 votes
2 answers
202 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 $x \in M$, ...
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5 votes
1 answer
177 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
113 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 ...
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6 votes
1 answer
161 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 ...
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6 votes
1 answer
195 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 $\...
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6 votes
1 answer
180 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 $...
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5 votes
1 answer
119 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
63 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 ...
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3 votes
1 answer
107 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 ...
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1 vote
1 answer
82 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 ...
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1 vote
1 answer
148 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
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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 $\...
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6 votes
1 answer
434 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/...
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1 vote
0 answers
96 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
206 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 ...
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1 vote
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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 ...
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5 votes
0 answers
153 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 $\geq 3$...
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9 votes
1 answer
474 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 $N$ ...
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5 votes
2 answers
675 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 ...
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2 votes
0 answers
115 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 $1$-...
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11 votes
0 answers
239 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
68 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
57 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)\...
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9 votes
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(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). ...
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5 votes
1 answer
209 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 ...
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13 votes
3 answers
913 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
897 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
297 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
140 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
70 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
89 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 ...
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4 votes
0 answers
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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 ...
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7 votes
0 answers
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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 ...
1 vote
2 answers
296 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 ...
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10 votes
2 answers
644 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 ...
10 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}...
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2 votes
1 answer
126 views

Just-not-nilpotent-by-compact quotient of a locally compact group

It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually ...
7 votes
2 answers
745 views

Quotients of finitely generated nilpotent groups

Is the following fact true? Let $N$ be a finitely generated nilpotent group, and denote its lower central series by $(N_r)_{r\ge 1}$, that is, $N_1=N$ and $N_{k+1}=[N,N_k]$ is the ...
4 votes
1 answer
304 views

Nilpotent of class 2 free product

Question. How is the nilpotent of class 2 (nil-2) free product of groups defined? I came across this construction reading the following paper. Alan H. Mekler (1981), Stability of nilpotent groups of ...
1 vote
1 answer
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Are Carter subgroups nilpotent projectors?

R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of $...
5 votes
1 answer
217 views

Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...
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1 vote
3 answers
262 views

p-groups and 2-generated abelian images

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
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2 votes
1 answer
816 views

Word metrics and finite index subgroups

Suppose that we are given some finitely generated group $ G $ and some finite index subgroup of it $ H $. Given a finite generating symmetric generating set $ S \subset G $, we can define the word ...
13 votes
1 answer
422 views

Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...
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1 vote
0 answers
124 views

Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$. Is ...
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