Geodesic causality in Kerr spacetimes with $\mathrm{\mid }a\mathrm{\mid }\ge M$

The analytic extension of the Kerr spacetimes into the negative radial region contains closed causal curves for any non-zero rotation parameter $a$ and mass parameter $M$. Furthermore, the spacetimes become totally vicious when $\mathrm{\mid }a\mathrm{\mid }>M$, meaning that through every point there exists a closed timelike curve. Despite this, we prove that the Kerr spacetimes do not admit any closed null geodesics when $\mathrm{\mid }a\mathrm{\mid }\ge M$. This result generalises recent findings by one of the authors, which showed the nonexistence of closed causal geodesics in the case $\mathrm{\mid }a\mathrm{\mid }<M$. Combining these results, we establish the absence of closed null geodesics in Kerr spacetimes for any non-zero $a$.