Geodesic
$k$-Free-like groups
Algebra i logika, Tome 48 (2009) no. 2, pp. 245-257

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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 $k\ge3$.
Keywords: $k$-free-like groups. 
@article{AL_2009_48_2_a4,
     author = {A. Yu. Olshanskii and M. V. Sapir},
     title = {$k${-Free-like} groups},
     journal = {Algebra i logika},
     pages = {245--257},
     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2009_48_2_a4/}
}
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A. Yu. Olshanskii; M. V. Sapir. $k$-Free-like groups. Algebra i logika, Tome 48 (2009) no. 2, pp. 245-257. http://geodesic.mathdoc.fr/item/AL_2009_48_2_a4/