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In fact, much more is true: if $H$ is a finitely generated subgroup of a one-relator group and $H$ has property (T) then $H$ is finite. Indeed, a classical theorem of Brodskii and Howie asserts that ...

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 "...

It is known that a right-angled Artin group $A(\Gamma)$ embeds in some one-relator group if and only if $\Gamma$ is a forest. In fact, any such $A(\Gamma)$ embeds in the one-relator group $\...