Suppose that $\langle a,b\rangle$, $\langle b,c\rangle$, and $\langle a,c\rangle$ are free, is it true then that $\langle a,b,c\rangle$ is a free group of rank 3? We assume that $a,b,c$ are distinct group elements.
I know this is false if only $\langle a,b\rangle$ and $\langle b,c\rangle$ are free, for example we can take $a=\sigma_1$, $b=\sigma_2$, and $c=\sigma_3$ in the braid group $B_n$ for $n>3$.
Hint: Consider a group with presentation: $$\langle a,b,c\mid abc=1\rangle$$
This group can be seen as the free group on any two elements of $a,b,c.$
If you want a non-free group, rather than a group which is not free of rank $3,$ it can be:
$$\langle a,b,c\mid (abc)^2=1\rangle$$