$k$-Free-like groups

The following results are proved.
In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are $k$-free-like for any $k\ge2$.
In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic $m$-generated group is $k$-free-like for every $k\ge m+1$ (resp., $k\ge m$).
Finally, Theorem 3 asserts that there exists a 2-generated periodic group $G$ which is $k$-free-like for every $k\ge3$.