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March 2021 On property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb{Z})$
Marek Kaluba, Dawid Kielak, Piotr W. Nowak
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Ann. of Math. (2) 193(2): 539-562 (March 2021). DOI: 10.4007/annals.2021.193.2.3

Abstract

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

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Marek Kaluba. Dawid Kielak. Piotr W. Nowak. "On property (T) for $\mathrm{Aut}(F_n)$ and $\mathrm{SL}_n(\mathbb{Z})$." Ann. of Math. (2) 193 (2) 539 - 562, March 2021. https://doi.org/10.4007/annals.2021.193.2.3

Information

Published: March 2021
First available in Project Euclid: 23 December 2021

Digital Object Identifier: 10.4007/annals.2021.193.2.3

Subjects:
Primary: 20F28 , 22D55

Keywords: automorphism groups of free groups , Kazhdan constants , Kazhdan's property (T) , spectral gaps

Rights: Copyright © 2021 Department of Mathematics, Princeton University

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Vol.193 • No. 2 • March 2021
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