# Do one-relator groups satisfy Haagerup property?

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?

• 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.
– YCor
Sep 13 '15 at 12:13
• @YCor, do you know if this holds for Baumslag's famous example $\langle a,b\mid a^{(a^b)}=a^2\rangle$?
– HJRW
Sep 17 '15 at 13:39
• @HJRW good question, I'll think about it.
– YCor
Sep 17 '15 at 21:46