Residually amenable groups I have two questions about residually amenable groups:

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*Is every finitely presented amenable group residually elementary amenable?

*Given $n$, is the free Burnside group of exponent $n$ on two generators residually amenable?

Regarding 1., I know that Grigorchuk constructed an example of a finitely presented amenable group that is not elementary amenable, but I was unsure if it could be residually elementary amenable.  Regarding 2., I believe it is an open question whether or not the free Burnside group on two generators is sofic, so if the answer to 2. is known, I'm guessing that it would be in the negative.
 A: (From my initial comments)

*

*No, there's a finitely presented ascending HNN extension of Grigorchuk's group, and it's an isolated group (it has a unique minimal nontrivial normal subgroup— more precisely every proper quotient is metabelian [Sapir-Wise]), so it's not residually elementary amenable. See details and references §5.7 in my 2007 J. Algebra paper with Guyot and Pitsch on isolated groups ArXiv link.


*As confirmed by Mark Sapir for each given exponent and number $\ge 2$ of generators the question of residual amenability of the given Burnside group is open, except in the few cases where it's known to be locally finite (exponent $\le 4$ and $6$).
A: Regarding "Regarding 1". (The) Grigorchuk group is residually finite. A non-residually finite amenable but not elementary amenable group was constructed by Anna Erschler in Not residually finite groups of intermediate growth, commensurability and non-geometricity. I do not know if her group is residually elementary amenable. Regarding 2: There is a conjecture due to Y. Shalom (see  The algebraization of Kazhdan's property (T). International Congress of Mathematicians. Vol. II, 1283–1310, Eur. Math. Soc., Zürich, 2006.) that free Burnside groups have Kazhdan Property (T). Since infinite finitely generated Burnside groups are not residually finite (Zelmanov), Shalom's conjecture would imply that infinite finitely generated Burnside groups are not residually amenable. The Shalom conjecture has been disproved in Osajda, Damian
Group cubization. With an appendix by Mikaël Pichot.
Duke Math. J. 167 (2018), no. 6, 1049–1055.
A: About 2), It is unknown if every bounded torsion amenable f.g. group is finite.
If that was true then for free Burnside groups residually amenable would be equivalent to residually finite. By Zelmanov's result, for large enough exponents, the free Burnside group with more than 1 generators is not residually finite.
