Santos Rego Y. Finiteness properties of split extensions of linear groups. Bielefeld: Universität Bielefeld; 2019.We investigate presentation problems for certain split extensions of discrete matrix groups. In the soluble front, we classify finitely presented Abels groups over arbitrary commutative rings R in terms of their ranks and the Borel subgroup of SL(2,R). In the classical set-up we prove that, under mild conditions, a parabolic subgroup of a classical group is relatively finitely presented with respect to its extended Levi factor. This yields, in particular, a partial classification of finitely presented S-arithmetic parabolic groups. Furthermore, we consider higher dimensional finiteness properties and establish an upper bound on the finiteness length of groups that admit certain representations with soluble image
