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

Abstract

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