Lower bounds for the top rational cohomology of arithmetic groups 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


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*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.
 A: Let $\mathcal{O}$ be the ring of integers in an algebraic number field $k$ and let $\text{cl}(\mathcal{O})$ be the class number of $\mathcal{O}$.  In my paper "Integrality in the Steinberg module and the top-dimensional cohomology of $\text{GL}_n(\mathcal{O}_k)$" (joint with Church and Farb), I prove the following two theorems:


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*In its vcd, the dimension of the rational cohomology group of $\text{SL}_n(\mathcal{O}_k)$ is at least $(\text{cl}(\mathcal{O})-1)^n$.  In particular, it is nonzero if $\text{cl}(\mathcal{O}) > 1$.

*Conversely, assume that $\text{cl}(\mathcal{O})=1$.  Furthermore, assume either that $k$ has a real embedding or that $\mathcal{O}$ is Euclidean.  Then in their vcd's the rational cohomology of $\text{SL}_n(\mathcal{O}_k)$ and $\text{GL}_n(\mathcal{O}_k)$ vanishes.  There is a similar vanishing result for the homology with certain twisted coefficients.
These are Theorems C and B of that paper (we remark that Theorem C also claims a similar bound for GL, but there is a subtle error in the proof for GL.  Our techniques give some lower bound, but we haven't pinned down precisely what it is.  That should be corrected soon.).
I remark that in the Euclidean case the vanishing result in 2 above is an old theorem of Lee and Szczarba from
R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), no. 1, 15-53.
For $n=2$, the lower bound in 1 above is pretty classical, but the vanishing result in 2 seems to be new (at least in the non-Euclidean case).  I'm not sure what happens for $S$-arithmetic groups.
The paper can be downloaded from my webpage here.
