FUNDAMENTA MATHEMATICAE 197 (2007) Polyhedra with virtually polycyclic fundamental groups have finite depth by Danuta Kolodziejczyk (Warszawa) Abstract. 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 con- nection with one of them, we proved that there exist polyhedra with polycyclic funda- mental 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 polyhe- dron. 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. 1. Introduction. In 1979, at the International Topological Conference in Moscow, K. Borsuk introduced the capacity and depth in the shape category of compacta, together with some relevant questions (see [B1]). (The basic notions and results of shape theory can be found in [B4], [DS], [MS].) Recall that a domination in a given category C is a morphism f : X → Y , X, Y ∈ Ob C , for which there exists a morphism g : Y → X of C such that fg = id Y . Then we say that Y is dominated by X , and we write Y ≤ X or X ≥ Y ; moreover, X<Y will denote that X ≤ Y holds but Y ≤ X fails. In the following, C is the homotopy category of CW-complexes and ho- motopy classes of cellular maps between them or the shape category of compacta (pointed or unpointed). 2000 Mathematics Subject Classification : 55P15, 55P55, 55P10. Key words and phrases : polyhedron, ANR, CW-complex, compactum, homotopy dom- ination, homotopy type, shape domination, shape, depth, index of h-proximity, simple constituent. Research partially supported by the Ministry of Sciences and Higher Education grant # 1 P03A 005 30. [229] c  Instytut Matematyczny PAN, 2007