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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
Full Text:
##### References:
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