Unbounded rank expanders, property (T), and upgrading

The problem of "unbounded rank expanders" asks 
whether we can endow a system of generators with a sequence of 
special linear groups whose degrees tend to infinity over quotient rings 
of Z such that the resulting Cayley graphs form an expander family.
Kassabov answered this question in the affirmative. Furthermore, the 
completely satisfactory solution to this question was given by 
Ershov and Jaikin--Zapirain (Invent. Math., 2010);  they proved
Kazhdan's property (T) for elementary groups over non-commutative 
rings. (T) is equivalent to the fixed point property with respect to 
actions on Hilbert spaces by isometries.

We provide a new framework to "upgrade" relative fixed point 
properties for small subgroups to the fixed point property for the 
whole group. It is inspired by work of Shalom (ICM, 2006). Our 
main criterion is stated only in terms of intrinsic group structure 
(but *without* employing any form of bounded generation). 
This, in particular, supplies a simpler (but not quantitative) 
alternative proof of the aforementioned result of Ershov and 
Jaikin--Zapirain.  

If time permits, we will discuss other applications of our result.