The study of $\Lambda$ polarization in Semi-Inclusive Deep Inelastic Scattering (SIDIS) with a transversely polarized target has often been suggested as a privileged way to access transversity, as the initial quark polarization can be transmitted to the hyperon and then observed via its weak decay. COMPASS measured the $\Lambda$ transversity-induced polarization as a function of Bjorken $x$, of the fraction $z$ of the longitudinal momentum of the fragmenting quark carried by the $\Lambda$ and of the $\Lambda$ transverse momentum $P_T$. The results are compatible with zero in all regions of phase space. Under the hypothesis that transversity is a valence object, the previous knowledge of the transversity function allows to get information on the ratio of polarized and unpolarized $z-$integrated $\Lambda$ fragmentation functions. On the other hand, if the $\Lambda$ polarization is entirely due to the $s$ quark, or if a quark-diquark model is applied, information can be obtained on the $s$ quark transversity function.