Two important tools in the study of totally disconnected locally compact groups, introduced by George Willis, are the scale function and tidy subgroups. In principle, these notions are well-defined also for discrete groups, but it seems that they have not been studied much. Is there a specific reason, why these tools are not interesting in the discrete setting?
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6$\begingroup$ Have you tried to specify these notions to discrete groups? the answer is pretty clear... (yes, they are well-defined for discrete groups, all the theory applies, not just "in principle"). $\endgroup$– YCorCommented Jul 13, 2022 at 8:37
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4$\begingroup$ To flesh out @YCor's comment: Willis's constructions involve looking at compact open subgroups, specifically those which form a base of neighbourhoods for the identity element. Now every discrete group has a rather well-behaved compact open subgroup which is a very good local approximation to the identity element ... $\endgroup$– Yemon ChoiCommented Jul 13, 2022 at 12:50
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One situation where you can apply tdlc group theory in a nontrivial way to discrete groups is if the group $G$ that you are interested in has a commensurated subgroup $H$ (meaning, all conjugates of $H$ differ from $G$ only by finite index). In that case the pair $(G,H)$ is sometimes known as a Hecke pair, and you can 'complete' the pair to a tdlc group $\hat{G}$, such that $H$ is the preimage of a compact open subgroup $\hat{H}$. Then you apply results about tdlc groups to $\hat{G}$, and deduce properties of how $H$ is conjugated inside $G$. The general idea was around before Willis's work in this area and used a number of times, but an example of where this method is used in conjunction with the scale/tidy theory specifically is in the following article of Shalom–Willis:
Y. Shalom and G. A. Willis, Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity, Geometric and Functional Analysis volume 23, pages 1631–1683 (2013)
https://link.springer.com/article/10.1007/s00039-013-0236-5
Another context where a nondiscrete tdlc group $H$ can tell you something about the discrete group $G$ is if $G$ sits inside $H$ as a lattice (something that will happen, for example, if $G$ acts properly cocompactly on a locally finite connected graph, and the automorphism group of that graph is nondiscrete). I don't know if this idea has been used that much yet specifically in conjunction with scale theory, but see for example this article:
G. A. Willis, The scale function and lattices, Proc. AMS Volume 145, Number 7, July 2017, Pages 3185–3190
