Is there a one-relator group with property (T)?


That is, is there an $n > 2$, and some $x \in F\_n$ (the free group on $n$ generators) such that the quotient of $F\_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?



Is there a one relator group with property (T)?
No, the abelianization of any such quotient will be infinite (the abelianization of $F\_n$ is $\mathbb{Z}^n$, which does not have a cyclic subgroup of finite index), but Kazhdan groups always have finite abelianization. 


