On property (T) for $\mathrm{Aut}(F_n)$ and ${\mathrm{SL}}_n(\mathbb{Z})$

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan’s property (T) for every $n⩾6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n⩾5$. We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n⩾6$) and of ${\mathrm{SL}}_n(\mathbb{Z})$ (with $n⩾3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n>6$.