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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1$\begingroup$ 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. $\endgroup$– YCorSep 13, 2015 at 12:13
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$\begingroup$ @YCor, do you know if this holds for Baumslag's famous example $\langle a,b\mid a^{(a^b)}=a^2\rangle$? $\endgroup$– HJRWSep 17, 2015 at 13:39
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1$\begingroup$ @HJRW good question, I'll think about it. $\endgroup$– YCorSep 17, 2015 at 21:46
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1 Answer
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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
answered Sep 12, 2015 at 18:48