Revistes Catalanes amb Accés Obert (RACO)

Non-free two-generator subgroups of SL$_2(\Bbb Q)$

S. P. Farbman


The question of whether two parabolic elements $A$, $B$ of SL$_{2}({\Bbb C})$ are a free basis for the group they generate is considered. Some known results are generalized, using the parameter $\tau = \text{tr}(AB) -2$. If $\tau = a/b \in {\Bbb Q}$, $| \tau | < 4$, and $|a| \leq 16$, then the group is not free. If the subgroup generated by $b$ in ${\Bbb Z}/a{\Bbb Z}$ has a set of representatives, each of which divides one of $b \pm 1$, then the subgroup of SL$_{2}({\Bbb C})$ will not be free.

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