The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

Quadratic algebra of square groups

Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.     

Non-Abelian tensor square and related constructions of p-groups

Archiv der Mathematik - Let G be a group. We denote by $$nu (G)$$ a certain extension of the non-Abelian tensor square $$[G,G^{varphi }]$$ by $$G times G$$. We prove that if G is a finite potent...     

Non-abelian GKM theory

We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.     

Economical adjunction of square roots to groups

How do we have to extend a group so that in the resulting group all elements of the original group be squares? We give a rather precise answer to this question (the best possible upper bound differs from our estimate by at most a factor of two) and pose several open questions.     

On the tensor square of non-abelian nilpotent finite-dimensional Lie algebras

For every finite p-group G of order p ⁿ with derived subgroup of order p ^m , Rocco [N.R. Rocco, On a construction related to the nonabelian tensor square of a group, Bol. Soc. Brasil. Mat. 1 (1991), pp. 63–79] proved that the order of tensor square of G is at most p ^n(n?m). This upper bound has been improved recently by the author [P. Niroomand, On the order of tensor square of non abelian prime power groups (submitted)]. The aim of this article is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given n-dimensional non-abelian nilpotent Lie algebra L with derived subalgebra of dimension m we have dim(L???L)?≤?(n???m)(n???1)?+?2. Furthermore for m?=?1, the explicit structure of L is given when the equality holds.     

Non-abelian differentiable gerbes

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid G-extensions, which we call “connections on gerbes”, and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for G-gerbes over stacks. We also introduce G-central extensions of groupoids, generalizing the standard groupoid S¹-central extensions. As an example, we apply our theory to study the differential geometry of G-gerbes over a manifold.     

Linear groups defined by decomposable tensor equalities

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.