  We construct and study finitely presented groups with quadratic Dehn function
(QD-groups) and present the following applications of the method developed in
our recent papers. (1) The isomorphism problem is undecidable in the class of
QD-groups. (2) For every recursive function $f$, there is a QD-group $G$
containing a finitely presented subgroup $H$ whose Dehn function grows faster
than $f$. (3) There exists a group with undecidable conjugacy problem but
decidable power conjugacy problem; this group is QD.
