Polyhedra with virtually polycyclic fundamental groups have finite depth
The notions of capacity and depth of compacta were introduced by K. Borsuk in the seventies together with some open questions. In a previous paper, in connection with one of them, we proved that there exist polyhedra with polycyclic fundamental groups and infinite capacity, i.e. dominating infinitely many different homotopy types (or equivalently, shapes). In this paper we show that every polyhedron with virtually polycyclic fundamental group has finite depth, i.e., there is a bound on the lengths of all descending sequences of different homotopy types (or shapes) dominated by this polyhedron. As a corollary, we deduce that for two ANR's with virtually polycyclic fundamental groups the so-called index of h-proximity, introduced by K. Borsuk in his monograph on retract theory, is finite. We also obtain an answer to some question of K. Borsuk concerning homotopy (or shape) decompositions of polyhedra into simple constituents.