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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?

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    $\begingroup$ Most 1-relator groups are known to be free-by-(virtually solvable) (which implies Haagerup). It sounds plausible to me that 1-relator groups all have this property. $\endgroup$
    – YCor
    Sep 13 '15 at 12:13
  • $\begingroup$ @YCor, do you know if this holds for Baumslag's famous example $\langle a,b\mid a^{(a^b)}=a^2\rangle$? $\endgroup$
    – HJRW
    Sep 17 '15 at 13:39
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    $\begingroup$ @HJRW good question, I'll think about it. $\endgroup$
    – YCor
    Sep 17 '15 at 21:46
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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

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