Thread by @PerfectoidPeter

*Question (research)

within loc compact T2 groups,

+ what makes Lie groups different to prof. groups is that they have a small neighbourhood from which every closed subgroup escapes

+ and (locally) prof. groups don't have such a neighbourhood for open subgroups

(1/x)

It's sad we have to require small subgroups but not for closed subgroups but for something even much stronger

open subgroups

but that also means that there is a structure in between? what is it like?

What are the locally compact Haussdorf groups G such that for every neighbourhood U of 1_G there is a closed nontrivial subgroup H \subseteq U...

...but still there is a neighbourhood U_0 of 1_G such that every open subgroup H of G escapes U_0?

#LieGroups #TopologicalGroups

#research #mathematics I haven't thought about this for more than 10 minutes yet, but clearly the discrete topology doesn't work, if someone comes up with a pretty explanation of what's in this valley between (connected?) Lie groups and locally profinite groups please DM me 🥰

I also think this NSS property is a clear obstacle for realizing Lie groups as (meaningful) inverse limits of topological groups, but maybe the other way around works?

What I mean is that maybe you can realize any (connected) Lie group as the closure of a direct limit of ...

...finite groups

(most basic example is S^1 is the closure of the roots of unity, which are an obvious direct limit of finite cyclic groups)

It would be great if such a general theorem existed, because it would also shed some light on this evil twin nature of Lie & prof groups

HERE ARE MORE EXPLICIT QUESTIONS:

1. Can you obtain any (connected?) Lie group as the closure of the direct limit of all its proper closed subgroups?

2. Can you always realize a (connected) Lie group G as the closure of HK for H and K proper closed subgroups of G?