Faddeev fixed center approximation to $\pi \bar{K} K^*$ system and the $\pi_1(1600)$

We investigate the three-body system of $\pi \bar{K} K^*$ by using the fixed-center approximation to the Faddeev equation, taking the interaction between $\pi$ and $\bar{K}$, $\pi$ and $K^*$, and $\bar{K}$ and $K^*$ from the chiral unitary approach. The study is made assuming scattering of a $\pi$ on a $\bar{K} K^*$ cluster, which is known to generate the $f_1(1285)$ state. The resonant structure around $1650$ MeV shows up in the modulus squared of the $\pi$-$(\bar{K} K^*)_{f_1(1285)}$ scattering amplitude and suggests that a $\pi$-$(\bar{K} K^*)_{f_1(1285)}$ state, with "exotic" quantum numbers $J^{PC} = 1^{-+}$, can be formed. This state can be identified as the observed $\pi_1(1600)$ resonance. We suggest that this is the origin of the present $\pi_1(1600)$ resonance and propose to look at the $\pi f_1(1285)$ mode in future experiments to clarify the issue.