Property (T) for $\mathrm{Aut}(F_n)$

I will present a recent proof, by Kaluba, Nowak, and myself, of the fact that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n>4$.  I will discuss how the same strategy gives a new proof of property (T) for $\mathrm{SL}_n(\mathbb Z)$.