Some prime factorization results for type II₁ factors

We prove several unique prime factorization results for tensor products of type II_1 factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if $R \otimes LF_{r_1} \otimes ... \otimes LF_{r_m}$ is isomorphic to a subfactor in $R \otimes LF_{s_1} \otimes >... \otimes LF_{s_n}$, for some $2\leq r_i, s_j \leq \infty$, then $m\le n$.