*[Corrected, thanks – T.]*

I mean the latter; I’ll reword to clarify. (The example of the solenoid group from Notes 0 shows that the former is not true.)

]]>Thank you for the reply. I don’t know your proof, but I am thinking about taking a group G which is not locally euclidean (say Q2 or nilpotent versions, easy to construct, or even some strange group a la Grigorchuk). Inside this group, maybe, you can choose a (sequence of?) approximate group(s) with two properties: first, the approximate group is big enough to feel (??) all the group G, second, you can tell exactly what is the locally euclidean group (say G’) which has inside a progression which is roughly equivalent with the approximate group.

The question is: is there any relation between G and G’? (like approximate morphism, or a functorial correspondence G-G’…)

I’m not sure what the “group of dyadic integers” is in Q2; also, to specify an approximate group, one needs to not just specify the ambient group G, but also a subset A of that group G obeying the approximate group axioms. So I don’t have enough information to answer your question.

If you mean G to be the additive group of rationals whose denominator is a power of two, and let be the multiples of between -1 and 1 for some large , then this is an approximate group, and in the limit , the associated ultraproduct can be modeled by the interval in the real line. But I don’t know if this is the type of example you have in mind.

]]>locally compact group?

*[Corrected, thanks – T.]*

1. Re Remark 5, I would say that, according to the definition of “finite dimensional” from Exercise 7, there is a lot of structure entering by the back door, namely something akin to rectifiability (statements about continuous images of euclidean spaces). Maybe related, the problem of defining rectifiability in sub-riemannian spaces (which are main examples of spaces non euclidean at any scale) is open, in the sense that there is no satisfactory definition of rectifiability for these spaces.

2. Question: In your Theorem 12 (Freiman’s type theorem) from your Notes 0, if you take to be the dyadic integers (or some noncommutative nilpotent like, but dyadic version), what could be the locally compact group generated by the said sequence of approximate groups, etc ?

Thanks!

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