The question of whether two parabolic elements A, B of SL2(C) are a free basis for the group they generate is considered. Some known results are generalized, using the parameter τ = tr(AB) - 2. If τ = a/b ∈ Q, |τ| < 4, and |a| ≤ 16, then the group is not free. If the subgroup generated by b in Z / aZ has a set of representatives, each of which divides one of b ± 1, then the subgroup of SL2(C) will not be free.
@article{urn:eudml:doc:41227,
title = {Non-free two-generator subgroups of SL2(Q).},
journal = {Publicacions Matem\`atiques},
volume = {39},
year = {1995},
pages = {379-391},
mrnumber = {MR1370894},
zbl = {0848.20040},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41227}
}
Farbman, S. Peter. Non-free two-generator subgroups of SL2(Q).. Publicacions Matemàtiques, Tome 39 (1995) pp. 379-391. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41227/