Basic Properties of \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\)

In the previous chapter, we have proved that every metrically completely reducible maximal irreducible solvable subgroup of \(\varDelta (V, \kappa )\) is conjugate to a subgroup of the form \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\) . Furthermore, we established some basic properties of groups of the form \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\) , such as their order. In this chapter, we will establish some further properties of groups of the form \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\) . For example, we will determine when \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\) is irreducible, and when \(G_{\mu ,\nu }^{\mathcal {B}}(X_1, \ldots , X_k)\) is irreducible and metrically primitive.