Related papers: Locally Nilpotent Groups and Hyperfinite Equivalen…

We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field…

We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary…

We show that if $G$ is a finitely generated group hyperbolic relative to a finite collection of subgroups $\mathcal{P}$, then the natural action of $G$ on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite…

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the…

Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative…

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We…

Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

We investigate the behavior of countable Borel equivalence relations (CBERs) on topological Ramsey spaces. First, we give a simple proof of the fact that every CBER on $[\mathbb{N}]^{\mathbb{N}}$ is hyperfinite on some set of the form…

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there…

We construct a finitely-presented group such that its Vogel-Levine localization is not transfinitely nilpotent. This answers a problem of J. P. Levine.

We prove that if a linear group $G$ is almost Engel, then $G$ is finite-by-hypercentral. If $G$ is almost nil, then $G$ is finite-by-nilpotent.

We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence…

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical…

This paper deals with the variety of commutative nonassociative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, $\gamma \in K$. Correa et al proved that if $\gamma = 0,1$ then any such finitely generated algebra is nilpotent.…

We prove a characterization of the amenability of countable Borel equivalence relations in terms of the uniform Liouville property for group actions on their classes. Furthermore, inspired by a well-known amenability criterion for locally…

We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most…

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated…

Here we show that a finite nilpotent group is 2-closed if and only if it is either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.

Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horv\'ath and Szab\'o; the…