Borel Complexity and the Schr\"oder-Bernstein Property

We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $\Phi$ of $\mathcal{L}_{\omega_1 \omega}$ and to every cardinal $\lambda$, the thickness $\tau(\Phi, \lambda)$ of $\Phi$ at $\lambda$. As applications, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete. We also show that under the existence of large cardinals, if $\Phi$ is a sentence of $\mathcal{L}_{\omega_1 \omega}$ with the Schr\"{o}der-Bernstein property (that is, whenever two countable models of $\Phi$ are biembeddable, then they are isomorphic), then $\Phi$ is not Borel complete.