4
$\begingroup$

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?

$\endgroup$
2
  • 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"). $\endgroup$
    – YCor
    Jul 13, 2022 at 8:37
  • 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 Choi
    Jul 13, 2022 at 12:50

1 Answer 1

2
$\begingroup$

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

https://www.jstor.org/stable/90006398

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.