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