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?

## 1 Answer

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.

Do simple discrete subgroups of Lie groups exist?Here's a good summary of current progress mathoverflow.net/q/90058/81055 $\endgroup$reallyabout 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. $\endgroup$