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For context, I'm only a second year undergraduate mathematician, so I won't know much.

For third year, I'm hoping to do a research project. I met up with a professor who might be my supervisor today, and he spoke about a certain result he had read about which I could do my project on. It was from a book/paper which had a model-theoretic construction for the Gromov boundary for a certain type of group (amenable groups or any group, I've forgotten).

We're trying to find this book/paper again. Would anyone here know of this result?

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    $\begingroup$ Why don't you ask your professor about details? Also, maybe your professor confused Gromov-boundary and the asymptotic cone... $\endgroup$
    – Moishe Kohan
    Commented May 20 at 17:31
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    $\begingroup$ Gromov's theorem on groups of polynomial growth and elementary logic, van den Dries, L.; Wilkie, A. J., J. Algebra 89 (1984), no. 2, 349–374. Sorry to say, but it does not sound like your professor knows enough of the relevant material to supervise student's research project on this subject... $\endgroup$
    – Moishe Kohan
    Commented May 20 at 17:39
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    $\begingroup$ @AlessandroCodenotti you probably mean arxiv.org/abs/math/0311101 Linus Kramer & Katrin Tent "Asymptotic cones and ultrapowers of Lie groups" $\endgroup$
    – Dmitrii Korshunov
    Commented May 20 at 18:56
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    $\begingroup$ @MoisheKohan He's just a busy guy, that's all. I'm sure he knows enough about the relevant material, he's just seen a lot of work. $\endgroup$
    – CatsAndDogs
    Commented May 20 at 20:56
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    $\begingroup$ I actually think this is a feasible problem. Non-principal ultrafilters are things that find convergent subsequences. Gromov boundaries are sets of limit points of geodesics modulo some equivalence relations. I think it could be interesting to construct Gromov boundaries using some kind of ultralimit. It may be "trivial" but definitely appropriate for an undergraduate project as it will require digesting two heavy duty definitions. All that said, amenability will play no role here. $\endgroup$
    – NWMT
    Commented May 21 at 12:38

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