ROOT-CLASS RESIDUALITY OF FUNDAMENTAL GROUP OF A FINITE GRAPH OF GROUP

Abstract

Let \(\mathcal{K}\) be an abstract class of groups. Suppose \(\mathcal{K}\) contains at least a non trivial group.
Then \(\mathcal{K}\) is called a root-class if the following conditions are satisfied:

1. If \(A \in \mathcal{K}\) and \(B \leq A\), then \(B \in \mathcal{K}\).

2. If \(A \in \mathcal{K}\) and \(B \in \mathcal{K}\), then \(A\times B \in \mathcal{K}\).

3. If \(1\leq C \leq B \leq A\) is a subnormal sequence and \(A/B, B/C \in \mathcal{K}\), then there exists a normal subgroup \(D\) in group \(A\)
such that \(D \leq C\) and \(A/D \in \mathcal{K}\).

Group \(G\) is root-class residual (or \(\mathcal{K}\)-residual), for a root-class \(\mathcal{K}\) if,
for every \(1 \not = g \in G\),
exists a homomorphism \(\varphi \) of group \(G\) onto a group of root-class \(\mathcal{K}\) such that \(g\varphi \not = 1\).
Equivalently, group \(G\) is \(\mathcal{K}\)-residual if, for every \(1 \not = g \in G\),
there exists a normal subgroup \(N\) of \(G\) such that \(G/N \in \mathcal{K}\) and \(g \not \in N\).
The most investigated residual properties of groups are finite groups residuality (residual finiteness),
\(p\)-finite groups residuality and soluble groups residuality.
All there three classes of groups are root-classes.
Therefore results about root-class residuality have safficiently enough general character.

Let \(\mathcal{K}\) be a root-class of finite groups.
And let \(G\) be a fundamental group of a finite graph of groups with finite edges groups.
The necessary and sufficient condition of virtual \(\mathcal{K}\)-residuality
for the group \(G\) is obtained.