Stability, non-approximated groups and high-dimensional expanders

Alexander Lubotzky examines whether infinite groups can be approximated by asymptotic homomorphisms into finite symmetric or unitary groups. He describes finitely presented groups that admit no unitary approximation in the Frobenius norm and several other norms. The argument links vanishing of higher cohomology to stability, then uses Garland’s method and high-dimensional expanders arising from Bruhat–Tits buildings. Certain non-residually finite central extensions of lattices in p-adic Lie groups are stable, preventing such approximations. These extensions come from a p-adic analogue of a result of Deligne. The talk explains the relevant notions and draws on joint work with M. De Chiffre, L. Glebsky, A. Thom and I. Oppenheim.