相关论文: The $R_\infty$ property for nilpotent quotients of…

The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a…

Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we…

A {\it graph product} $G$ on a graph $\Gamma$ is a group defined as follows: For each vertex $v$ of $\Gamma$ there is a corresponding non-trivial group $G_v$. The group $G$ is the quotient of the free product of the $G_v$ by the commutation…

This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact K\"ahler manifold) is a solvable group of matrices over Q (respectively…

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

This work introduces and investigates the function $J(G) = \frac{\text{Nil}(G)}{L(G)}$, where $\text{Nil}(G)$ denotes the number of nilpotent subgroups and $L(G)$ the total number of subgroups of a finite group $G$. The function $J(G)$,…

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime…

Let K be an algebraically closed field. For a graded K-Algebra R, we write cmdef R:=dim R -depth R. We show that for each reductive group G (over K) which is not linearly reductive, there exists a faithful G-module V such that cmdef…

Let $\{ G_n\}_{n\in\w}$ be a closed tower of metrizable groups. Under a mild condition called $(GC)$ and which is strictly weaker than $PTA$ condition introduced in [22], we show that: (1) the inductive limit…

Given a smooth, irreducible, projective surface $S$, let $g(S)$ be the minimum geometric genus of an irreducible curve that moves in a linear system of positive dimension on $S$. We determine the value of this birational invariant for a…

We consider conformal actions of solvable Lie groups on closed Lorentzian manifolds. With anterior results in which we addressed similar questions for semi-simple Lie group actions, this work contributes to the understanding of the identity…

Let $G$ be a finite group with the property that if $a,b$ are powers of $\delta_1^*$-commutators such that $(|a|,|b|)=1$, then $|ab|=|a||b|$. We show that $\gamma_{\infty}(G)$ is nilpotent.

Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a…

Given a group word $w$ in $k$ variables, a finite group $G$ and $g\in G$, we consider the number $N_{w,G}(g)$ of $k$-tuples $g_1,\dots ,g_k$ of elements of $G$ such that $w(g_1,\dots ,g_k)=g$. In this work we study the functions $N_{w,G}$…

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The…

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…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…