Amenable automatic groups

Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if a group is both amenable and automatic? Two of the biggest open questions about Thompson's group $$F$$ are whether it is amenable, and whether it is automatic, and I was wondering whether there's at least some established reason that it can't be both.

• I suspect the answer is no. Note that (finitely generated) solvable groups are not automatic except when they are virtually abelian. May 24 at 11:41
• The solvable Baumslag-Solitar groups ${\rm BS}(1,n)$ are however asynchronously automatic (meaning that the multiplier automata can read their two input tapes at different rates), while not being automatic. May 24 at 12:40
• Most amenable but not elementary amenable groups I know are not finitely presented. But I'm no expert May 24 at 12:47
• @Zaremsky If you can show automaticity of the bigger group would give the Grigorchuk group a rational crosssection you could eliminate that possibility. Or the Dehn function is likely to big. May 24 at 18:50
• The Tits alternative for automatic groups seems to be an open problem. (See eg here: automaticgroups.tumblr.com.) Your question is very similar, and so is surely open too.
– HJRW
May 25 at 7:49