Certain residual properties of HNN-extensions with central associated subgroups

Certain residual properties of generalized Baumslag–Solitar groups

“…The use of the concept of a root class turned out to be very productive in the study of some residual properties of free constructions of groups: generalized free and tree products, HNN-extensions, fundamental groups of graphs of groups, etc. It allows one to prove several statements at once and quickly complicate the constructions under consideration; see, for example, [40,[42][43][44][45][46][47][48]. At the same time, if C is a root class different from the class of all finite groups, very few facts are known about the conjugacy C-separability of free constructions of groups.…”

On the separability of subgroups of nilpotent groups by root classes of groups

“…The notion of a root class was introduced in [10] and allows one to prove many statements at once using the same reasoning. It turns out to be especially useful in studying the residual properties of free constructions of groups (see, e.g., [22][23][24][25][26][27][28][29][30][31]). If X is such a construction and C is a root class of groups, then the C-separability of some subgroups of X is quite often one of the necessary and/or sufficient conditions for X to be residually a C-group.…”

On the separability of subgroups of nilpotent groups by root classes of groups

“…At the same time, the approximability by other classes of groups is also considered in the literature, and many of these classes are root classes of groups.In accordance with one of the equivalent definitions (see Proposition 3.2 below), a class of groups C is called a root class if it contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian products of the form y∈Y X y , where X, Y ∈ C and X y is an isomorphic copy of X for each y ∈ Y . The concept of a root class was introduced by K. Gruenberg [5] and turned out to be very useful in studying the approximability of the fundamental groups of various graphs of groups [1,4,[13][14][15][16][17][18]21,22]. Thanks to its use, it became possible, in particular, to make significant progress in the study of the residual p-finiteness (where p is a prime number) and the residual solvability of such groups.Everywhere below, it is assumed that Γ = (V, E) is a non-empty connected undirected graph with a vertex set V and an edge set E (loops and multiple edges are allowed).…”