TITLE: Statements in group theory which imply deep results in number theory
QUESTION [21 upvotes]: Can we name some examples of theorems in group theory which imply (in a relatively straight-forward way) interesting theorems or phenomena in number theory?
Here are two examples I thought of:
The existence of Golod-Shafarevich towers of Hilbert class fields follows from an inequality on the dimensions of the first two cohomology groups of the ground field.
Iwasawa's theorem on the size of the $p$ part of the class groups in $\mathbb{Z}_p$-extensions follows from studying the structure of $\mathbb{Z}_p[\![T]\!]$-modules.
Can you name some others?

REPLY [4 votes]: This is not a straightforward example, but the Oppenheim conjecture was proved originally by Margulis using ergodic theory and group theory.

REPLY [4 votes]: A conjecture was made by Dunfield and Calegari that certain congruence covers of an arithmetic hyperbolic 3-manifold have trivial first betti number (which corresponds to the non-existence of certain automorphic forms, conjectured based on the generalized Riemann hypothesis and the Langlands proram). This was subsequently proved by Boston and Ellenberg using methods from pro-$p$ groups.