Questions tagged [nilpotent-groups]

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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 ...
Alexander Chervov's user avatar
2 votes
0 answers
124 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}\...
Christopher-Lloyd Simon's user avatar
2 votes
1 answer
155 views

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 ...
Corentin B's user avatar
10 votes
1 answer
544 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 ...
Sean Eberhard's user avatar
0 votes
1 answer
158 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 ...
enihcamemit's user avatar
6 votes
1 answer
225 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 ...
Kyle's user avatar
  • 243
2 votes
0 answers
70 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 $\gamma_{k+1}(F_n)$ ...
Arthur's user avatar
  • 21
4 votes
0 answers
193 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 ...
semisimpleton's user avatar
5 votes
0 answers
344 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|}\...
Sebastien Palcoux's user avatar
3 votes
1 answer
113 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 ...
Zach Hunter's user avatar
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3 votes
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138 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 ...
Qwert Otto's user avatar
0 votes
0 answers
158 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, ...
mick's user avatar
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1 answer
91 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/...
Surajit's user avatar
  • 73
1 vote
0 answers
58 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 $\...
abracadabra12345's user avatar
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}(...
Thiago Luiz's user avatar
3 votes
0 answers
59 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\} \...
Petr Naryshkin's user avatar
3 votes
0 answers
90 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\...
Meisam Soleimani Malekan's user avatar
5 votes
0 answers
151 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 ...
Sean Lawton's user avatar
  • 8,384
9 votes
2 answers
228 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 $x \in M$, ...
Alice's user avatar
  • 93
5 votes
1 answer
207 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 $< p$,...
Joshua Grochow's user avatar
7 votes
1 answer
173 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 ...
Roberta's user avatar
  • 153
6 votes
1 answer
227 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 ...
Irina's user avatar
  • 437
7 votes
1 answer
269 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 $\...
Irina's user avatar
  • 437
6 votes
1 answer
234 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 $...
Irina's user avatar
  • 437
5 votes
1 answer
169 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 $...
Christian Gorski's user avatar
0 votes
0 answers
70 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 ...
Totoro's user avatar
  • 2,515
3 votes
1 answer
133 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 ...
ARG's user avatar
  • 4,342
1 vote
1 answer
178 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^{}}}) ...
HIMANSHU's user avatar
  • 381
4 votes
0 answers
132 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 $\...
Anton Mellit's user avatar
  • 3,572
6 votes
1 answer
568 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/...
user avatar
1 vote
0 answers
124 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 $$ \{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G $$ ...
Niall Taggart's user avatar
3 votes
1 answer
232 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 ...
Houa's user avatar
  • 561
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 ...
ARG's user avatar
  • 4,342
5 votes
0 answers
170 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 $\geq 3$...
M. Dus's user avatar
  • 1,900
9 votes
1 answer
599 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 $N$ ...
J. Darné's user avatar
  • 263
5 votes
2 answers
922 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 ...
Tom1990's user avatar
  • 51
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 $1$-...
M. Dus's user avatar
  • 1,900
11 votes
0 answers
248 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 \...
Jakub Konieczny's user avatar
2 votes
1 answer
77 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 ...
Sven Wirsing's user avatar
2 votes
0 answers
67 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)\...
Alex Doe's user avatar
  • 205
9 votes
0 answers
425 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). ...
Denis T's user avatar
  • 4,371
5 votes
1 answer
249 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 ...
Ivan's user avatar
  • 445
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} = ...
Nate Eldredge's user avatar
25 votes
0 answers
959 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 ...
David E Speyer's user avatar
1 vote
1 answer
352 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 $...
Jiayin Pan's user avatar
1 vote
0 answers
143 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} =...
Omar Antolín-Camarena's user avatar
2 votes
0 answers
73 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 ...
Changguang's user avatar
1 vote
1 answer
99 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 ...
Mikhail Katz's user avatar
  • 15.1k
4 votes
0 answers
136 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 ...
thibo's user avatar
  • 333
7 votes
0 answers
420 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 ...
Grisha Papayanov's user avatar