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 $\mathrm{\Phi }$ of $$\mathcal{L}$_{\omega_1 \omega}$andtoeverycardinal$\lambda$,thethickness$\tau(\Phi, \lambda)$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 $\mathrm{\Phi }$ is a sentence of ${\mathcal{L}}_{{\omega }_1\omega }$ with the Schr"{o}der-Bernstein property (that is, whenever two countable models of $\mathrm{\Phi }$ are biembeddable, then they are isomorphic), then $\mathrm{\Phi }$ is not Borel complete.