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I remember reading several times that Lie groups, and particularly compact Lie groups can be seen as analogous to finitely generated groups.

I can see the analogy between the generators, exponential map and so on but I don't know any formalization of those analogies.

Is there some theory unifying Lie groups and finitely generated groups ?

Shaun
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Weier
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  • Can you link to a specific source that says this? – Qiaochu Yuan Nov 21 '20 at 20:31
  • I can't find an "authoritative" source but the OP in https://math.stackexchange.com/questions/3368390/developing-intuition-for-lie-groups-and-lie-algebras is expressing the same idea regarding exponential, generators, etc. – Weier Nov 21 '20 at 21:17
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    There's one, which is "locally compact groups", and particularly compactly generated ones, which for discrete groups means finitely generated groups. If you're interested in harmonic analysis, this analogy is classical. In geometric group theory, it's a leitmotiv for the book Metric geometry of locally compact groups by myself and Pierre de la Harpe – YCor Nov 23 '20 at 16:21

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