Kazhdan constants for Chevalley groups over the integers
Abstract: N. Ozawa showed that for every group with property (T) there exists a sum-of-squares decomposition of Δ2 - λΔ in RG. This means that for every group property (T) can be proven by performing a single computation. In 2021 we (with D. Kielak and P. Nowak) showed that for a whole family of groups SL(n,Z), n≥3, there exists a single element Adj3 in RSL(3,Z) such that property (T) for the whole family can be established by exhibiting a sum of squares decomposition of Adj3. As a byproduct the computation allowed us to establish the best known lower bounds for the Kazhdan constant κ(En, SL(n,Z)) with n≥5.
In this talk I will focus on extending those results beyond the case of SL(n,Z) to groups graded by root systems. I will show how to connect the analogous elements in the group rings (called the adjacency operators AdjΦ, to the root system Φ of the underlying Chevalley groups. As an outcome, we (with D. Kielak) were able to give the very first lower bounds for the universal Chevalley groups over Z corresponding to all irreducible root systems of rank at least 2. In particular, we give the first lower bounds for Kazhdan constants for the symplectic groups Sp(2n,Z), with respect to the usual (Steinberg) generators.