Over the ring of integers, groups of type Φ were first introduced by Olympia Talelli as a possible algebraic characterisation of groups that admit finite dimensional models for classifying spaces for proper actions. In this short article, we make the same definition over arbitrary commutative rings of finite global dimension and prove a number of properties pertaining to cohomological invariants of these groups with the extra condition that the groups belong to a large hierarchy of groups introduced by Peter Kropholler in the nineties. We prove most of Talelli's conjecture of equivalent statements for type Φ groups for these groups, and expand the scope of a few existing results in the literature.

The author was supported by a research scholarship from the Department of Mathematics, University of Manchester where they undertook this research as a graduate student. During the time period of making revisions to the original draft, the author was supported by an Early Career Research Fellowship, Grant no. ECF 1920-64, of the London Mathematical Society. They also express their thanks to their Ph.D. supervisor Peter Symonds for many useful discussions, and to the anonymous referee for their invaluable insights and comments.