Interesting Theorems on Finitely Generated Abelian Groups?

Question

First time teaching algebraic topology, probably gonna be related to most of my questions on here for a while.

I was wondering if anyone knows of particularly interesting theorems or examples from the theory of finitely generated abelian groups, beyond the standard auxiliary stuff and the classification/primary decomposition. By auxiliary I mean stuff like the basic results on subgroups, quotients, torsion/rank uniqueness and change of base. Doesn't have to be something that can quickly be PROVEN from first principles, but as long as it's something digestible and believable (or even better, unbelievable). Perhaps certain actions on manifolds or trees or the like? Something for topology-minded folks rather than algebra-minded would be ideal, but whatever you think is cool I'd love to hear about!

It could also be surprising internal/structural features, or results about their automorphism groups.

Thanks, hope someone finds this topic interesting!

Answer 1

Somewhat related: All subgroups of finitely generated abelian groups are finitely generated, however this is not necessarily so for subgroups of finitely generated groups.

Perhaps the classic example of the free group of 2 generators has subgroups that are not finitely generated, and its relation to algebraic topology: Commutator subgroup of rank-2 free group is not finitely generated.

Answer 2

For every finite generating set $S$ of $G=Z^n$, the generating function of its growth $$ \sum_{n=0}^\infty \beta_S(n)t^n $$ is rational. Here $\beta_S(n)$ is the number of group elements $g\in G$ of norm $|g|\le n$ and $|g|$ is the distance from $g$ to $0$ in the Cayley graph of $G$ associated with $S$.

This also holds for virtually abelian groups (i.e. groups containing $Z^n$ as a finite index subgroup) and may other groups. However, surprisingly, rationality becomes tricky for nilpotent groups: It depends on the generating set!

Benson, M., Growth series of finite extensions of $Z^n$ are rational, Invent. Math. 73, 251-269 (1983). ZBL0498.20022.

Stoll, M., Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126, No. 1, 85-109 (1996). ZBL0869.20018.

Answer 3

1. A Theorem of Gromov states that: If $G$ is quasi-isometric to $\mathbb{Z}^{n}$ then $G$ has a finite index subgroup isomorphic to $\mathbb{Z}^{n}$. (From a geometric information for the group we get algebraic information.) Therefore $\mathbb{Z}^{n}$ is quasi-isometrically rigid. The proof for general $n$ is quite difficult but for $n=1$ is far easier.
2. Also an interesting topic is the mapping class group $Mod(M)$ (the group of symmetries of a topological surface $M$). If $M$ is compact and orientable then $Mod(M)$ is generated by Dehn twists. Now two examples are if $M$ is the annulus $S^{1}\times[0,1]$ then $Mod(M)\cong \mathbb{Z}$ and if $M$ is a pair of pants (a sphere with three disks removed) then $Mod(M)\cong \mathbb{Z}^{3}$.