Related papers: Nilpotent probability of compact groups

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

Let $G= \exp(\g)$ be a connected, simply connected nilpotent Lie group. We show that for every $G$-invariant smooth sub-manifold $M$ of $g^*$, there exists an open relatively compact subset $\mathcal{M}$ of $M$ such that for any smooth…

Given a profinite group $G$ and a family $\mathcal{F}$ of finite groups closed under taking subgroups, direct products and quotients, denote by $\mathcal{F}(G)$ the set of elements $g \in G$ such that $\{x \in G\ |\ \langle g,x \rangle \…

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $\epsilon > 0$ such that if G is a residually finite group with…

We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic…

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner…

We prove a sufficient condition for a finite clique complex to collapse to a $k$-dimensional complex, and use this to exhibit thresholds for $(k+1)$-collapsibility in a sparse random clique complex. In particular, if every strongly…

Let $ n, q $ be positive integers. We show that if $ G $ is a finitely generated residually finite group satisfying the identity $ [x,_ny^q]\equiv 1, $ then there exists a function $ f(n) $ such that $ G $ has a nilpotent subgroup of finite…

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth…

Let G be a profinite group. The following results are proved. The commutator subgroup G' is finite if and only if G is covered by countably many abelian subgroups. The group G is finite-by-nilpotent if and only if G is covered by countably…

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique…

The semigroup of the homotopy classes of the self-homotopy maps of a finite complex which induce the trivial homomorphism on homotopy groups is nilpotent. We determine the nilpotency of these semigroups of compact Lie groups and finite Hopf…

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the…

For a finite group $G$, we study the probability $sp(G)$ that, given two elements $x,y \in G$, the cyclic subgroup $\langle x \rangle$ is subnormal in the subgroup $\langle x, y \rangle$. This can be seen as an intermediate invariant…

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper,…

Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group. In this work we show that $G$ embeds into the group of orientation preserving $C^{1+\alpha}$-diffeomorphisms of the compact interval, for all $\alpha< 1/k$ where…

We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the…

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting…

Let k be an algebraically closed field of positive characteristic and G a simple algebraic group defined over k. Under the assumption that the characteristic is a good prime for G, we determine a maximal G-stable subvariety U' of the…