Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

On virtual cabling and a structure of 4-strand virtual pure braid group

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extension and find presentation of [Formula: see text] and [Formula: see text]. As an application to classical braids, we find a new presentation of the Artin pure braid group [Formula: see text] in terms of the cabled generators.

On 3-strand singular pure braid group

In this paper, we study the singular pure braid group [Formula: see text] for [Formula: see text]. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that [Formula: see text] is a semi-direct product [Formula: see text], where [Formula: see text] is an HNN-extension with base group [Formula: see text] and cyclic associated subgroups. We prove that the center [Formula: see text] of [Formula: see text] is a direct factor in [Formula: see text].

On the profinite topology on solvable groups

We show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

A small infinitely-ended 2-knot group

We show that a 2-knot group discovered in the course of a census of 4-manifolds with small triangulations is an HNN extension with finite base and proper associated subgroups, and has the smallest base among such knot groups.