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](http://arxiv.org/abs/1204.2132)'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?



An infinite, amenable, finitely presentable group with no non-trivial finite quotients
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. 


