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Is there a scaling-limit theory for groups the way there is for graphs ("graphons") and permutations ("permutons")?

E.g., if we map the uniform measure on the cyclic group $\mathbf{Z}/n\mathbf{Z}$ to uniform measure on the set $\{0/n,1/n,...,(n-1)/n\}$, then we can represent the group law for $\mathbf{Z}/n\mathbf{Z}$ as a discrete measure on $[0,1]^3$ satisfying various group-y properties, and these measures weakly converge to a continuous measure that represents (the graph of) the group law for $\mathbf{R}/\mathbf{Z}$.

Of course the limit objects would have to be called "groupons".

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  • $\begingroup$ Pretty different from what you're asking, but profinite groups are kinds of limits of finite groups: en.wikipedia.org/wiki/Profinite_group $\endgroup$
    – Sam Hopkins
    Commented Sep 22, 2021 at 2:47
  • $\begingroup$ Ultralimits are not enough? $\endgroup$
    – markvs
    Commented Sep 22, 2021 at 3:52
  • $\begingroup$ Direct and inverse limits are important concepts but I'm looking for something more flexible that captures some of the latent geometry of the group. Meanwhile ultra limits seem TOO floppy! Maybe I should ask a specific question: Is there a version of what I said above about R/Z (which also works for (R/Z)^n) and applies it to the group of Mobius transformations vis-a-vis its discrete subgroups? $\endgroup$
    – James Propp
    Commented Sep 24, 2021 at 18:49
  • $\begingroup$ What is TOO floppy about ultralimits? $\endgroup$
    – markvs
    Commented Sep 25, 2021 at 13:43
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    $\begingroup$ I think the following paper implies that in this restrictive sense you don't get much more than your simple example : zbmath.org/?q=an%3A1328.54020 (i did not check it in detail but it seems interesting in relation with your question anyway). $\endgroup$
    – Jean Raimbault
    Commented Sep 29, 2021 at 6:23

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