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My question is a simple one: is there a group with the properties in the title?

In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a finitely generated, infinite, simple, amenable group. On the other hand, all the standard examples that I know of finitely presentable groups without finite quotients seem to be non-amenable.

There are a number of participants on MO who have constructed interesting examples of amenable groups. My guess is that the answer to this question is unknown, but if so I'd also be interested in informed conjectures. Is it at all probable that such a group doesn't exist?

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    $\begingroup$ It's a classical open question. The f.g. case was also open until the 2012 examples of Juschenko-Monod (for which finite generation and non-existence of finite quotients was established by Matui). $\endgroup$ – YCor Jan 29 '14 at 18:12
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    $\begingroup$ I'd certainly conjecture that such groups DO exist. $\endgroup$ – YCor Jan 29 '14 at 18:12
  • $\begingroup$ Thanks for both of these comments, Yves. Is the question written down anywhere? The Kourovka notebook, perhaps? $\endgroup$ – HJRW Jan 29 '14 at 19:31
  • $\begingroup$ I'm not sure it's written this way. Basically it stands at the middle between two classical questions (1) find a f.g. infinite amenable group with no nontrivial quotient (2) find a f.p. infinite amenable simple group. Now (1) is solved but the examples are hopelessly infinitely presentable. $\endgroup$ – YCor Jan 29 '14 at 20:10
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    $\begingroup$ @HJRW: oh, I was't thinking clearly when I asked about virtually solvable case. More to the point, Osin proved in Proposition 3.1 of arxiv.org/abs/math/0404075 that every f.g. elementary amenable group has a virtually polycyclic quotient, and hence has a nontrivial finite quotient. $\endgroup$ – Igor Belegradek Jan 29 '14 at 23:24
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It's a classical open question. The finitely generated case was also open until the 2012 examples of Juschenko-Monod (for which finite generation and non-existence of finite quotients was established by Matui 2006). Matui also checked that these examples are not finitely presented.

Such groups can certainly not be elementary amenable, since it is an elementary exercise to show that an infinite f.g. elementary amenable group always admits an infinite virtually abelian quotient and hence admits finite quotients of unbounded cardinal.

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