  In this paper we analyze the asymptotic behavior of the Dirichlet fractional
Laplacian $(-\Delta_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded
domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions.
A dimension reduction phenomenon is observed and described via
$\Gamma$-convergence.
