Related papers: On co-Hopfian nilpotent groups

We investigate the commuting automorphisms of nilpotent Lie algebras $L$ with coclass $\leq 3$. Our examination exposes the conditions under which the set of commuting automorphisms of $L$ forms a subgroup within its automorphism group.

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair…

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually…

We show that certain cyclically pinched one-relator groups are residually torsion-free nilpotent.

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on…

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group…

We develop nilpotently $p$-localization of knot groups in terms of the (symplectic) automorphism groups of free nilpotent groups. We show that any map from the set of conjugacy classes of the outer automorphism groups yields a knot…

We consider the so-called {\it strongly co-Hopfian} and {\it uniformly strongly co-Hopfian} Abelian groups, significantly generalizing some important results due to Abdelalim in the J. Math. Analysis (2015). Specifically, we prove that any…

We describe the structure of "K-approximate subgroups'' of solvable subgroups of GL_n(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free…

We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that…

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

We establish vanishing results for the first cohomology group of nilpotent groups and Lie rings when the submodule of invariants is trivial. Our results are obtained within a model-theoretic setting, namely for structures that are definable…

In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie…

The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the…

We construct finitely generated simple torsion-free groups with strong homological control. Our main result is that every subset of $\mathbb{N} \cup \{\infty\}$, with some obvious exceptions, can be realized as the set of dimensions of…

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$.…

A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to…

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group…

We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate…

It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup…