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On property (T) for \(\mathrm{Aut}(F_n)\) and \(\mathrm{SL}_n(\mathbb{Z})\). (English) Zbl 07331715
Summary: 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 \(n>6\).

MSC:
22D55 Kazhdan’s property (T), the Haagerup property, and generalizations
20F28 Automorphism groups of groups
Software:
SCS; GAP; Julia ; JuMP; Hecke; Nemo
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References:
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