Yes.

]]>That is correct. But there are many LERF groups that are not 3-manifold groups.

]]>Exactly!

]]>It doesn’t follow from Moise. I think you’re assuming that any subclass of a recursively enumerable class is recursively enumerable – but this isn’t true. For instance, the class of all finitely presented groups is recursively enumerable, for silly reasons!

But there are plenty of classes of finitely presentable groups that are not recursively enumerable. In a comment above, I said that I couldn’t think of any, but I was overlooking the fact that the argument in the RECURSIVENESS section shows that the class of all ‘not 3-manifold’ groups is not recursively enumerable! Similarly, if P is any recursively enumerable Markov property then the class of groups with not P is not recursively enumerable.

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