While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical tools (beside their presence in the definitions). The author often uses various decompositions such as Iwasawa and Bruhat, Levi and Langlands decomposition of parabolic subgroups, Hermite and Smith normal forms and other straightforward matrix calculations. But I don't see if this works the other way around. The question is:

What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about the structure of $SL(n,\mathbb{Z})$?

The question is motivated by the fact that from the viewpoint of structure theory of algebraic groups $SL_n,~n\geqslant3$ behaves much better than $SL_2$, while modular forms for $SL_2$ seem to be studied more that those for groups of higher ranks. So let me formulate another version of the above question (not sure if it is less vague):

In what aspects, if any, automorphic forms for $SL(n,\mathbb{Z}),~n\geqslant3$ behave better than those for $SL(2,\mathbb{Z})$?

I apologize, if all of the above is trivial (or nonsense), but I am as far from being an expert in modular forms as possible.

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