2-dimensional polyhedra with finite depth

In this paper every polyhedron is finite and every ANR is compact. Let P≥X1≥X2≥…, be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer lP (independent of the sequence) that each sequence contains only ≤lP homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra{Pi}, wherePihomotopy dominatesPi+1butPiandPi+1have different homotopy types, for everyi∈N? (M. Moron) [30, Problem 1436], and the famous problem of K. Borsuk (1967): Is it true that twoANR′shomotopy dominating each other have the same homotopy type? Here we prove that, if dim⁡P=2, then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim⁡P≤2 and elementary amenable fundamental group G with cdG<∞, there is a bound lP on the lengths of all descending sequences P≥X1≥X2… of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth.