On root-class residuality of generalized free products

Let $G$ be a generalized Baumslag-Solitar group and $\mathcal{C}$ be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for every $y \in Y$. We give a criterion for $G$ to be residually a $\mathcal{C}$-group provided $\mathcal{C}$ consists only of periodic groups. We also prove that $G$ is residua...

It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is ...

A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two constructions, namely amalgamated products with cyclic amalgams and cyclic HNN extensions.

Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.

A finitely generated residually finite group $G$ is an $\widehat{OE}$-group if any action of its profinite completion $\widehat G$ on a profinite tree with finite edge stabilizers admits a global fixed point. In this paper, we study the profinite genus of free products $G_1*_HG_2$ of $\widehat{OE}$-groups $G_1,G_2$ with finite amalgamation $H$. Given such $G_1,G_2,H$ we give precise formulas for the number of isomorphism classes of $G_1*_HG_2$ and of its profinite completion....

Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X...

Let R be a class of groups closed under taking semidirect products with finite kernel and fully residually R-groups. We prove that R contains all R-by-{finitely generated residually finite} groups. It follows that a semidirect product of a finitely generated residually finite group with a surjunctive group is surjunctive. This remained unknown even for direct products of a surjunctive group with the integers Z.

This paper is devoted to the proof of the property of order separability for free product of free groups with maximal cyclic amalgamated subgroups.

In this paper we investigate the properties of quasipotency and $g$-potency for free products with amalgamation.

Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian powers of a certain type). It is known that $\mathfrak{G}$ is residually a $\mathcal{C}$-group if it has a homomorphism onto a group from $\mathcal{C}$ acting injectively on all v...