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A famous theorem of Borel and Serre says that if $R$ is the ring of integers in an algebraic number field, then $\text{SL}_n(R)$ satisfies virtual Bieri-Eckmann duality. In other words, there exists a torsion-free finite-index subgroup $\Gamma$ of $\text{SL}_n(R)$ and a $\text{SL}_n(R)$-module $D$ such that

$$H^{\nu - k}(\Gamma;M) \cong H_{k}(\Gamma;M \otimes D)$$

for all $k$ and all $\text{SL}_n(R)$-modules $M$, where $\nu < \infty$ is the cohomological dimension of $\Gamma$.

Question : Are there any other rings $R$ such that $\text{SL}_n(R)$ is known to be a virtual duality group? Or where it is known not to be one? I suppose that one necessary condition is that $\text{SL}_n(R)$ has to have finite virtual cohomological dimension, so I'm only interested in counterexamples in which that holds (so, for instance, I'm not interested in $\text{SL}_n(\mathbb{C})$).

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1 Answer 1

Actually, Borel and Serre proved the result you mention for S-arithmetic groups: If $K$ is a number field, $S$ a finite set of places of $K$ and $\mathcal{O}_S$ the subring of $K$ of elements that are integral outside $S$ (called $S$-integers) then $SL_n(\mathcal{O}_S)$ is a virtual duality group. This is shown in

Borel, Serre: Cohomologie d'immeubles et de groupes S-arithmétiques. Topology 15(1976), 211-232. Théorème 6.2

To give an example, if $K=\mathbb{Q}$ then $\mathcal{O}_S=\mathbb{Z}[\frac{1}{p_1},...,\frac{1}{p_s}]$ where $p_i$ are rational primes.

Of interest may also be a result of Behr in the functional field case: Let $K$ be a finite extension of the function field $\mathbb{F}_{p^n}(X)$. Then, with the same notation as above, $SL_n(\mathcal{O}_S)$ has a duality subgroup $\Gamma_0$ of finite index. However, in contrast to the number field case, $\Gamma_0$ has $p$-torsion. (see Theorem 1 of this paper).

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