  Kazhdan's property (T) has been studied for several discrete group-like
structures, including standard invariants of Jones' subfactors and discrete
quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in
this setting. This allows us to construct a family of property (T) discrete
quantum groups, as well as subfactor standard invariants, given by generators
and a triangle presentation. These are the first examples of property (T)
discrete quantum groups that are not directly related to a discrete group with
property (T).
