I would like to know what estimates exist for the dimension of $H^d({\rm GL}_2(\mathcal{O}_{K,S}),\mathbb{Q})$ where $\mathcal{O}_{K,S}$ is a ring of $S$-integers in a number field $K$ and $d$ is the virtual cohomological dimension of the group ${\rm GL}_2(\mathcal{O}_{K,S})$. As the title says, I am mostly interested in lower bounds or asymptotic statements measuring the growth of the top cohomology in terms of arithmetic information (like genus, discriminant of $K$ or some such number , and the number of places in $S$). Moreover, information on the possible methods to obtain such estimates (from Euler characteristic computations, via study of Hecke operators, representation theory or so) would be very welcome.

An example of a formula in the spirit of the question is the estimate of the cuspidal cohomology of Bianchi groups in

- J. Rohlfs. On the cuspidal cohomology of the Bianchi modular groups. Math. Z. 188 (1985), no. 2, 253–269.

I'm wondering if there are similar statements for general rings of $S$-integers. I haven't even been able to find asymptotics for the rational cohomology of ${\rm GL}_2(\mathbb{Z}[1/n])$ for $n\to \infty$ in the literature.

Maybe one more point describing my motivation: in the case of a function field of a curve $C$ over a finite field, the situation is easier to understand. The top rational cohomology can be computed in terms of Gauss-Bonnet formulas and is then related to the zeta-value $\zeta_C(-1)$ for the curve. Estimates for the rational cohomology then follow from the Weil conjectures (so the growth is roughly $q^{2g+s-3}$ with $g$ the genus of the curve, $s$ the number of places, and $q$ the size of the ground field). Essentially I want to know if a similar picture exists (possibly conjecturally?) on the number field side.