Does there exist a finitely generated discrete group $G$ such that it has property (T), but for every $\varepsilon > 0$ there exists a generating set $S$ with the corresponding Kazhdan constant less than $\varepsilon$?



Uniform bounds on Kazhdan constants in groups
The answer is yes, by Gelander and Zuk: <http://www.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=697297&vfpref=html&r=24&mx-pid=1910934>


