It is well-known that SL(n,Z) for n at least 3 has Kazhdan's property (T). By the Delorme--Guichardet theorem, this property is equivalent to saying that every group action on a Hilbert space by affine isometries admits a global fixed point.
First proof of this (due to Kazhdan) exploited the fact that SL(n,Z) is a lattice in SL(n,R). Later, Shalom found a way to bypass this fact by appealing to another fact that SL(n,Z), n at least 3, is Boundedly Generated ("BG") by elementary matrices. However, this BG condition itself is super-strong, and may not be desired in more general situation.
In this talk, I will present the most "down-to-earth" proof: it does not employ either of lattice structure or any form of BG. That proof enables us to prove fixed point properties for a much wider class of groups, along exactly the same line.