Radicals and Plotkin's problem concerning geometrically equivalent groups

Hirsch-Plotkin radical of stability groups

We study the Hirsch-Plotkin radical of stability groups of (general) subspace series of infinite dimensional vector spaces. We show that in countable dimension and some other cases, the HP-radical of the stability group coincides with the set of all space automorphisms that fix a finite sub-series; this implies that the HP radical is a Fitting group. Conversely, we prove that every countable Fitting group, which is either torsion-free or a p-group may be represented as a subgroup of the HP radical of a series stabilizer.

Geometric equivalence of $π$-torsion free nilpotent groups

In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin \cite{P1, P2}. Sufficient and necessary conditions are presented for a $π$-torsion-free nilpotent group to be geometrically equivalent to its $π$-completion. We prove that a relatively free nilpotent $π$-torsion-free group and its $π$-completion define the same quasi-variety. Examples of $π$-torsion-free nilpotent groups that are geometrically equivalent to their $π$-completions are given.

On groups in which subnormal subgroups of infinite rank are commensurable with some normal subgroup

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that the intersection of H and N has finite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has infinite rank or G is nilpotent-by-abelian (and has infinite rank).

Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.

Invariant random subgroups of groups acting on hyperbolic spaces

Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix any point of $\partial S$. We prove that every invariant random subgroup of $G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $(X,μ)$ either has finite stabilizers…

代数群单模的几何刚性

Geometric rigidity of simple modules for algebraic groups

研究代数群单模的几何刚性，证明所有单\(G -\)模是几何刚性但非绝对刚性，通过研究特定代数找到与单模\(V\)相关的有限纯不可分扩张\(k_V/k\)，并在\(G\)光滑连通时结合相关理论给出\(k_V\)具体描述及细化维数公式等。