A math thread! There is a beautiful conjecture about the web of normal subgroups inside mapping class groups of finite-type surfaces: https://twitter.com/littmath/status/1353169427848716289

1. Any injective map of a geometric* subgroup H into its corresponding mcg must be the identity map twisted by an automorphism of H.

* If the automorphism group of a normal subgroup is the full (extended) mapping class group, let’s call it a “geometric” subgroup.

This would mean that geometric subgroups are wholly unique among all subgroups! No subgroup is abstractly isomorphic to H except for H itself. A genuine mapping class group snowflake.

(Stunningly, it is unknown if this is true even for the Torelli subgroup!)

Now, that’s fun and good. However ... do geometric subgroups exist? If they do, how common are they? How can you tell if any particular given normal subgroup is geometric? This leads me to the second conjecture:

2. A normal subgroup is geometric if and only if it is not isomorphic to a right-angled Artin group.

This is extremely descriptive. RAAGs have a very particular isomorphism type. This conjecture says that normal subgroup come in exactly two flavors:

A one-of-a-kind snowflake, or a right-angled Artin group.

I find this profoundly beautiful. The structure of mapping class groups is so rich that every normal subgroup can either remember the full structure of the group — or has a narrow, rigid Artin structure.

Wonderfully, there is serious evidence that this ought to be true. Geometric subgroups are known to be wholly unique ... in the class of their peers, other normal subgroups. Moreover, every non-geometric subgroup found has been a RAAG — and infinitely many of them at that!

P.S. I contend that conjectures like these ought to be among the ways to truly “understand” an infinite group, such as how groups are understood now by their presentations, representations, naming schemes, actions, and/or intrinsic metric data.