Induction of spectral gaps for the cohomological Laplacians of SL(n,ℤ) and SAut(F_n).

Abstract

Bader and Nowak introduced cohomological Laplacians whose properties are linked to vanishing of group cohomologies. In particular, for a group $G$, the degree one Laplacian possessing a positive spectral gap is equivalent to $G$ satisfying Kazhdan's property (T). The latter statement is a consequence of the works of Bader, Nowak and Sauer. On the other hand, due to Ozawa, it had been known before that property (T) is equivalent to the existence of a positive spectral gap related to the degree zero Laplacian. Kaluba, Kielak and Nowak applied the characterization of Ozawa to show property (T) for the automorphism groups of free groups. They developed an induction technique for that purpose - from the existence of a spetral gap for the automorphism group of smaller degree, they were able to deduce the spectral gap for higher degree automorphism groups. In this talk, we describe a method of adapting the induction technique of Kaluba, Kielak and Nowak to the degree one Laplacian. As an aplication of our method, we provide an alternative proof of property (T) for the group of $n\times n$ integer matrices with determinant one for $n\geq 3$.