TITLE: Do one-relator groups satisfy Haagerup property?
QUESTION [6 upvotes]: The question is in the title:

Do one-relator groups satisfy Haagerup property?

I think the answer is known at least in some specific cases, but is the problem completely solved?

REPLY [2 votes]: As far as I know, the problem is not solved, but much is known. The canonical reference is
the suggestively named Groups with Haagerup property: Gromov's a-T-menability, by Cherix, Cowling, Jolissant, Julig, Valette. See, in particular, Theorem 6.3.1 and Chapter 7