Centraliser dimension and universal classes of groups

In this paper we establish results that will be required for the study of the algebraic geometry of partially commutative groups. We define classes of groups axiomatized by sentences determined by a graph. Among the classes which arise this way we find $\mathrm{CSA}$ and $\mathrm{CT}$ groups. We study the centralisers of a group, with particular attention to the height of the lattice of centralisers, which we call the centraliser dimension of the group. The behaviour of centraliser dimension under several common group operations is described. Groups with centraliser dimension $2$ are studied in detail. It is shown that $\mathrm{CT}$-groups are precisely those with centraliser dimension $2$ and trivial centre.