For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points, then the open curve $C=\overline{C}\setminus\{P_1,\dots,P_n\}$ is hyperbolic if $2-2g-\sum \deg P_i<0$.

The general function-field/number-field analogy suggests the following question:

Is there a notion of hyperbolicity for rings of $S$-integers in number fields (possibly along the lines of some inequality appearing in arithmetic intersection theory)?

If there is such a notion of hyperbolicity, are most $S$-integer rings hyperbolic? Is it also true that number rings with trivial or finite étale fundamental group (as in this MO-question) are not hyperbolic?

Here is some of the background which motivated the question: In the function field case, I think I have convinced myself that the slightly stronger inequality $2-2g-n<-1$ implies that there will be cuspidal cohomology for the group $SL_2(k[C])$ (in the sense that the quotient of the product of Bruhat-Tits trees for the points $P_1,\dots,P_n$ is rationally non-contractible). I am trying to understand if similar statements can be made in the number-field situation. Therefore, the ideal answer to the question would be some numerical inequality which relates to Harder's Gauß-Bonnet formula for cohomology of arithmetic groups, implying that hyperbolicity forces non-trivial rational cohomology for the arithmetic group $SL_2(\mathcal{O}_{K,S})$.

For algebraic curves over a nice enough field $k$, we have a notion of what it means to be hyperbolic: If $\overline{C}$ is a smooth projective curve of genus $g$ and $P_1,\dots,P_n$ are closed points, then the open curve $C=\overline{C}\setminus\{P_1,\dots,P_n\}$ is hyperbolic if $2-2g-\sum \deg P_i<0$.

The general function-field/number-field analogy suggests the following question:

Is there a notion of hyperbolicity for rings of $S$-integers in number fields (possibly along the lines of some inequality appearing in arithmetic intersection theory)?

If there is such a notion of hyperbolicity, are most $S$-integer rings hyperbolic? Is it also true that number rings with trivial or finite étale fundamental group (as in this MO-question) are not hyperbolic?

Here is some of the background which motivated the question: In the function field case, I think I have convinced myself that the slightly stronger inequality $2-2g-n<-1$ implies that there will be cuspidal cohomology for the group $SL_2(k[C])$ (in the sense that the quotient of the product of Bruhat-Tits trees for the points $P_1,\dots,P_n$ is rationally non-contractible). I am trying to understand if similar statements can be made in the number-field situation. Therefore, the ideal answer to the question would be some numerical inequality which relates to Harder's Gauß-Bonnet formula for cohomology of arithmetic groups, implying that hyperbolicity forces non-trivial rational cohomology for the arithmetic group $SL_2(\mathcal{O}_{K,S})$.