Prove these are not surface groups

For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$For $n = 1$, these are the fundamental groups of close genus $g$ surfaces. I expect that they are not isomorphic to fundamental groups of closed surfaces for $n \geq 2$, but I can't figure out a proof (except for the trivial case $g=1$, where they have torsion). Note that from the abelianization you can see that if they are surface groups they are also genus $g$ surface groups. Can anyone prove this?