Current research areas of Group Theory

Question

I'm finishing an undergraduate degree in mathematics and am really passionate about group theory. Currently, my research projects have been mostly in ring theory, however (since my university does not have any group theorists).

Thus, I really don't know how to go about getting to know more research level problems in group theory - the broad areas, the big open questions, etc. I'm afraid of committing a graduate degree to it only to end up regretting it later.

I took a look at ArXiv's "Group Theory" section, but it didn't really help me get the big picture, especially when it comes to algebraic aspects of the theory (many papers were a lot more geometric in nature, which does not please me much, personally).

In short, what are the current big areas of group theory? And what problems do they try to answer?

Thanks in advance!

Answer 1

If you don't like geometry, maybe group theory is not for you. In its origin, group theory is closely related to how they act on spaces. Group action is a powerful tool for studying both the space and the group. And it's still the case today.

Geometric group theory is a relatively new and extremely active research area, which can no longer be distinguished from combinatorial group theory. Even problems of purely algebraic/discrete/computational nature, such as the Burnside problem or the decidability of the word problem can be illuminated by (continuous) geometry. There are tons of unsolved problems in this area. This branch is also closely related to topology, operator algebras, probability and ergodic theory, etc, where virtually Haken conjecture (solved) and Baum-Connes conjecture (widely open) are among the most important things.

Algebraic group theory is also an active research area, which should be considered as part of algebraic geometry and Lie theory. The structure, rational points and representations of algebraic groups are closely related to problems from number theory, in particular the Langlands program. But this is rather involved, would require even more ring theory and is seldom considered as part of "group theory". However, this type of mathematics is often argubly more "algebraic", if that's the most important thing for you.

All said, group (theory) is ubiquitous in modern math. There is hardly any math that's either pure or free of groups. You cannot have too much preconception about what you like until you try.

Answer 2

Group theory is part of many research areas, as was already mentioned. If you prefer algebraic methods, then algebraic groups, Lie algebras, representation theory and number theory are active research areas related to group theory.

For groups and Lie algebras, the Lazard correspondence gives a link between $p$-groups and Lie rings and Lie algebras. Zelmanov's work on the restricted Burnside problem for groups uses a lot of Lie algebra and Jordan algebra theory, for example.

For groups and number theory, there are many links. To give just one example, group cohomology is used in number theory a lot, e.g., Galois cohomology. But there are many more links. And as an example of a famous result, the Oppenheim conjecture was proved originally by Margulis using ergodic theory and group theory.

Answer 3

Big picture of group theory, try
https://mathscinet.ams.org/mathscinet/freeTools.html?version=2
choose "Search MSC", then
20 Group Theory and Generalizations