  We look at constructions of aperiodic SFTs on fundamental groups of graph of
groups. In particular we prove that all generalized Baumslag-Solitar groups
(GBS) admit a strongly aperiodic SFT. Our proof is based on a structural
theorem by Whyte and on two constructions of strongly aperiodic SFTs on
$\mathbb{F}_n\times \mathbb{Z}$ and $BS(m,n)$ of our own. Our two constructions
rely on a path-folding technique that lifts an SFT on $\mathbb{Z}^2$ inside an
SFT on $\mathbb{F}_n\times \mathbb{Z}$ or an SFT on the hyperbolic plane inside
an SFT on $BS(m,n)$. In the case of $\mathbb{F}_n\times \mathbb{Z}$ the path
folding technique also preserves minimality, so that we get minimal strongly
aperiodic SFTs on unimodular GBS groups.
