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We consider a natural question: "Is it true that each homotopy domination ofa polyhedron over itself is a homotopy equivalence?" and a strongly relatedproblem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominatingeach other have the same homotopy type?" The answer was earlier known to bepositive for manifolds (Bernstein-Ganea, 1959), $1$-dimensional polyhedra andpolyhedra with polycyclic-by-finite fundamental groups (DK, 2005). Thus one mayask, if there exists a counterexample among $2$-dimensional polyhedra withsoluble fundamental groups. In this paper we show that it cannot be found inthe class of $2$-dimensional polyhedra with soluble fundamental groups $G$ withcd$G \leq 2$ (and soluble can be replaced here by a wider class of elementaryamenable groups). We prove more general fact, that there are no counterexamplesamong $2$-dimensional polyhedra, whose fundamental groups have finiteaspherical presentations and are Hopfian (or more general, weakly Hopfian). Inparticular, a counterexample does not exist also among $2$-dimensionalpolyhedra whose fundamental groups are knot groups and in the class of$2$-dimensional polyhedra with one-related torsion-free Hopfian fundamentalgroups. The results can be applied also, for example, to hyperbolic groups orlimit groups with finite aspherical presentations.  For the same classes of polyhedra we get also a positive answer to anotherother open question: "Are the homotopy types of two quasi-homeomorphic ANR'sequal?"

Polyhedra for which every homotopy domination over itself is a homotopy equivalence