Related papers: Intersections of Magnus subgroups and embedding th…
A group G is said to have the Magnus property if the following holds: whenever two elements x,y have the same normal closure, then x is conjugate to y or its inverse. We prove: Let p be an odd prime, and let G,H be residually finite-p…
Free products of two residually finite groups with amalgamated retracts are considered. It is proved that a cyclic subgroup of such a group is not finitely separable if, and only if, it is conjugated with a subgroup of a free factor which…
A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible $A_1$ subgroups of exceptional algebraic groups $G$. Consequences are given…
We show that the Magnus embedding, which embeds the free solvable group of rank r and degree d into the wreath product of the free abelian group of rank r with the free solvable group of rank r and degree d-1, is a quasi-isometry.
We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.
We describe all reductive spherical subgroups of the group SL(n) which have connected intersection with any parabolic subgroup of the group SL(n). This condition guarantees that any open equivariant embedding of the corresponding…
A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under…
A conjecture of Dicks and the author on rank of the intersection of factor-free subgroups in free products of groups is proved for the case of left ordered groups.
We find a condition on the acylindrical action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first…
The statement ``no nonabelian simple group can be obtained from a nonsimple group by adding one generator and one relator" 1) is equivalent to the Kervaire--Laudenbach conjecture; 2) becomes true under the additional assumption that the…
For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does…
We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of…
An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups. However, not much is…
We construct an efficient model for graphs of finitely generated subgroups of free groups. Using this we give a very short proof of Dicks's reformulation of the strengthened Hanna Neumann Conjecture as the Amalgamated Graph Conjecture. In…
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite…
We present obstruction results for self-similar groups regarding the generation of free groups. As a main consequence of our main results, we solve an open problem posed by Grigorchuk by showing that in an automaton group where a…
We study a class $\mathfrak{M}$ of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic…
A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question…
Magnus' Freiheitssatz states that if a group is defined by a presentation with $m$ generators and a single relator containing the last generating letter, then the first $m-1$ letters freely generate a free subgroup. We study an analogue of…
A group $G$ is said to be intersection-saturated if for every strictly positive integer $n$ and every map $c\colon \mathcal{P}(\{1,\dots, n\})\setminus \emptyset \rightarrow \{0,1\}$, one can find subgroups $H_1,\dots, H_n\leq G$ such that…