# Questions tagged [nilpotent-groups]

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### 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 ...
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### 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 ...
117 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 ...
369 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 ...
294 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 ...
550 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 ...
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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 $\langle N,S\... 3answers 246 views ### p-groups and 2-generated abelian images Let$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? 1answer 790 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 $, we can define the word ... 1answer 414 views ### Variety of nilpotent Lie algebras or$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}$... 0answers 111 views ### Lower central series in a free pro-p group Let$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]$. Is ... 0answers 218 views ### A normal form theorem for presentations of finite$p$-groups of nilpotency class$2$? When constructing examples of nonabelian finite$p$-groups with abelian automorphism group (and certain other desired properties), the authors of papers like http://arxiv.org/pdf/1304.1974v1.pdf leave ... 1answer 233 views ### Subgroups of Nilpotent groups with prescribed center Let$G$be a torsion-free, finitely-generated, nilpotent group of nilpotency class at least 3. Does there exist a normal subgroup$N\leq G$such that$G/N\cong \mathbb{Z}$and$Z(G)=Z(N)$? (By$Z(H)...
Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies: $$A n^d \leq B_n \leq Bn^d$$ for some constants $A$, $B$. My question ...