Related papers: On universal and epi-universal locally nilpotent g…

For any compact Lie group $G$ and any $n$ we construct a smooth $G$-manifold $U_n(G)$ such that any smooth $n$-dimensional $G$-manifold can be embedded in $U_n(G)$ with a trivial normal bundle. Furthermore, we show that such embeddings are…

A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.

We develop a method to show that some (abstract) groups can be embedded into a free pro-$p$ group. In particular, we show that a finitely generated subgroup of a free $\mathbb Q$-group can be embedded into a free pro-$p$ group for almost…

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality…

This paper aims to investigate the self-similarity property in finitely-generated torsion-free nilpotent groups. We establish connections between geometric equivalence and self-similarity in these groups. Moreover, we show that any…

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of $G$-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the…

The class of those (torsion-free) $SI^*$-groups which are not locally soluble, has the cardinality of the continuum. Moreover, these groups are not only pairwise non-isomorphic, but also they generate pairwise different varieties of groups.…

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $\lambda$ such that there is a universal abelian $p$-group for purity of cardinality $\lambda$, i.e., an abelian $p$-group $U_\lambda$ of cardinality $\lambda$ such that every…

For cardinals lambda, kappa, theta we consider the class of graphs of cardinality lambda which has no subgraph which is (kappa, theta)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak)…

A group is called $\Lambda$-free if it has a free Lyndon length function in an ordered abelian group $\Lambda$, which is equivalent to having a free isometric action on a $\Lambda$-tree. A group has a regular free length function in…

Assume V=L and lambda is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext(G,Z) (under well known limitations), we construct an abelian group G of…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…

Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question…

We give an affrmative answer to the question whether a residually finite Engel group satisfying an identity is locally nilpotent. More generally, for a residually finite group G with an identity, we prove that the set of right Engel…

We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH…

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional…

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an…

In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a…

Let $k$ be an algebraically closed field of any characteristic except 2, and let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$ we…