TITLE: Is there a one relator group with property (T)?
QUESTION [8 upvotes]: 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 $\mathrm{(T)}$ ?

REPLY [3 votes]: Property (T) implies Property FA: every action on a tree has a global fixpoint. The Magnus–Moldavanskii hierarchy expresses every one-relator group as (a subgroup of) an HNN extension of a "simpler" one-relator group (over a free subgroup, but that doesn't matter here). The hierarchy terminates in finitely many steps with a free group or finite cyclic group. Putting this together, any group with (T) that embeds in a one-relator group fixes a vertex of the Bass–Serre tree, so is a subgroup of a simpler one-relator group, and acts on the Bass–Serre tree of that one-relator group, etc until we conclude that, since it can't be a non-trivial subgroup of a free group (infinite abelianization), it must be finite.