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

TL;DR

This paper establishes Kazhdan's property (T) for automorphism groups of free groups and special linear groups over integers for various n, providing explicit bounds and extending known results.

Contribution

It proves property (T) for Aut(F_n) for n ≥ 6, extending to n ≥ 5 with prior work, and offers explicit Kazhdan constant bounds for SAut(F_n) and SL_n(ℤ).

Findings

Property (T) holds for Aut(F_n) when n ≥ 6.

Explicit lower bounds for Kazhdan constants are provided.

Bounds improve previous results for n > 6 in SL_n(ℤ).

Abstract

We prove that $\operatorname{Aut}(F_n)$ 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 $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.