On property (T) for SL(n,Z) and Aut(F_n)
Let G_n be either SL(n,Z) or Aut(F_n). I will show how the action of outer automorphism groups provides control over embeddings of group rings R[Gn]→R[Gm](for mn). By exercising this control one can derive sum of squares decompositions of specific elements in R[Gm](for all m) from a single decomposition in R[Gn]. This reduces the proof of property (T) for Gm(for all m) to a single computation in Gn(n=3 in case of G=SL(n,Z), or n=5 in case of G=Aut(F_n)).
By successfully performing the calculation in arXiv:1812:03456, we provide a uniform argument, that Gn has property (T), obtaining additionally asymptotically optimal bounds on so called Kazhdan constants. This talk is based on joint work with Dawid Kielak (Oxford) and Piotr Nowak (IMPAN).