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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 $\varepsilon$?

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    $\begingroup$ It seems so, by Gelander and Zuk ams.org/mathscinet/search/… $\endgroup$
    – Dan Sălăjan
    Commented Nov 20, 2013 at 8:31
  • $\begingroup$ Yes, "most" higher rank lattices have this property. $\endgroup$
    – Misha
    Commented Nov 20, 2013 at 13:52
  • $\begingroup$ @Dan: thanks - can you give this comment as an answer so that I can accept it? $\endgroup$
    – Michal Kotowski
    Commented Nov 20, 2013 at 14:22

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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

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