I was given the problem above, and I'd appreciate some help. I think I have a general direction, but I'm not entirely sure if what I'm doing is true, so it'd be great if someone could tell me if what I'm doing is generally alright and maybe point me in a more concrete direction if necessary.
So I start off by computing the fundamental group of the genus $g$ handlebody. I do so by taking the handlebody and making it thinner and thinner, until its basically as thick as a line. This is a deformational retract, and therefore not supposed to affect the fundamental group. I'm then left with g copies of $S^1$ which all intersect at a point, and so by the Seifert-Van-Kampen theorem, the fundamental group is the free group generated by $g$ elements.
Now, I want to see what happens when I glue two handlebodies along the boundary. This is where things get tricky for me. I want to use Seifert-Van-Kampen once again, but not entirely sure that I'm doing it correctly. Since I glue the to handlebodies along the boundary, the intersection of the two handlebodies is now a sufrace of genus $g$, who's fundamental group I can once again calculate using Seifert-Van-Kampen -
$$\pi=\langle a_1,b_1,...,a_g,b_g| \prod _{i=1}^{g} [a_i,b_i]=1 \rangle$$
So I think that the fundamental group I'm looking for should be the free product of the two original fundamental groups of the handlebodies ($F_1, F_2$), under quotient with respect to the fundamental group of the surface with genus $g$ :
$$\pi'=(F_1*F_2)/\pi$$
Does this make sense? Or is there something I'm missing along the way? Moreover, I'm wondering if there's a simpler/clearer way to express the group $\pi'$.
Thanks in advance.