I know that the theory of Lie groups is a very old subject, and the literature is incredibly vast, so I am wondering what contemporary research on Lie groups is about. What open problems are there in this subject and is there any low-hanging fruit?
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1$\begingroup$ In addition to my answer, I know that there are (potentially) much more accessible conjectures in area of approximation theory: which groups are hyperlinear? how different pseudoinvariant metrics on Lie groups behave under homomorphisms? (I'd suggest to look at talks/papers by A. Lubotzky for context and precise statements). Also there are some problems regarding images of word maps and their action on Haar measure, connected to (pro)finite group theory; for me this is mostly hearsay, so I hope that somebody can elaborate on those topics. $\endgroup$– Denis TCommented Aug 24, 2023 at 15:04
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1$\begingroup$ And another folklore one, which is probably not very important, but very much open and intriguing. Do simple discrete subgroups of Lie groups exist? Here's a good summary of current progress mathoverflow.net/q/90058/81055 $\endgroup$– Denis TCommented Aug 24, 2023 at 15:19
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3$\begingroup$ I think the Hilbert--Smith conjecture is a very famous and old problem about Lie groups: en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture $\endgroup$– Sam HopkinsCommented Aug 24, 2023 at 15:25
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2$\begingroup$ @SamHopkins I was about to write this as well, but after a second thought I'd say that Smith conjecture is not really about Lie groups; similarly to 5th Hilbert problem, Gleason theorem, Hofmann-Morris theorem and such, it's about unexpected niceness of fluid and wild general topological groups, rather than some immanent (...maybe it is better to say "natural and intrinsically definable") properties of very algebraic, very rigid Lie groups. $\endgroup$– Denis TCommented Aug 24, 2023 at 16:53
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In my opinion, the most important and (immediately) widely applicable open problem in Lie group theory is Milnor isomorphism conjecture.
Let $G$ be a Lie group (not necessarily simple, although the statement is reduced to the case of simple groups). Consider a change of topology map $\operatorname{disc}: G^{\delta} \to G$, which is identity on elements, and source is the underlying group with discrete topology.
Friedlander-Milnor conjecture. $H^*(\operatorname{disc}, \Bbb Z/p)$ is an isomorphism.
Currently this is known to hold for solvable groups (not hard to prove), $H^2$ of semisimple algebraic groups, few low-dimensional examples (result for $H_3(SL_2)$ is by Sah, maybe there was some progress on recent years for third cohomology of some other small rank groups), and for $GL_{\infty}(k)$, $O_{\infty}(k)$, and $Sp_{\infty}(k)$, where $k$ is either $\Bbb R$ or $\Bbb C$. (If you don't like to think of those as Lie groups, you can restate this as isomorphism conjecture being true for usual linear groups, but only for dimensions in the stable range.)
Why is this problem important? It reduces (practically unfeasible) computation of discrete cohomology to a well-known object. Discrete cohomology serves as the target of various characteristic classes (i. e. works of Haefliger on classification of foliations), residue maps in algebraic K-theory, and allows us to understand flat bundles much better, possibly obtaining good generalisations of Milnor-Wood inequality.
