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Zbl 0770.11028
Yang, Jun
On the real cohomology of arithmetic groups and the rank conjecture for number fields. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 3, 287-306 (1992). ISSN 0012-9593

Let $G$ be a connected semisimple algebraic group defined over $\bbfQ$, and let $\Gamma\subset G$ be an arithmetic subgroup. Write $X$ for the symmetric space of $G(\bbfR)$ with respect to a maximal compact subgroup $K$ of $G$. One has a homomorphism $j\sp*:I\sp*\sb G\to H\sp*(\Gamma)$, where $I\sp*\sb G$ and $H\sp*(\Gamma)$ denote the group of $G$-invariant forms on $X$ and the real cohomology of $\Gamma$, respectively. By work of Garland, Hsiang and Borel one knows that $j\sp q$ is surjective in dimensions not greater than a constant $m(G)$, and Borel proved that $j\sp q$ is also injective in dimensions not greater than a constant $c(G/\bbfQ)$. Here Borel's result is improved to prove the rank conjecture for algebraic number fields (of degree $\ge 2$). For any commutative ring one has the rank filtration $r\sb \bullet$ of the rational $K$-groups $K\sb n(R)\sb \bbfQ=K\sb n(R)\otimes\bbfQ$ defined by $$r\sb iK\sb n(R)\sb \bbfQ=\{\text{Im}:H\sb n(GL\sb i(R);\bbfQ)\to H\sb n(GL(R);\bbfQ)\}\cap PH\sb n(GL(R);\bbfQ),$$ where $PH$ denotes the group of primitive homology classes. One also has the $\gamma$-filtration $\gamma\sp \bullet$ on $K\sb n(R)\sb \bbfQ$. The rank conjecture claims that $K\sb n(R)\sb \bbfQ=r\sb iK\sb n(R)\sb \bbfQ\oplus\gamma\sp{i+1}K\sb n(R)\sb \bbfQ$. The main theorem of the underlying paper says that $j\sp q$ is injective for $q\le l(G/\bbfQ):=2c(G/\bbfQ)+1$, where $l(G/\bbfQ)$ is a constant that can be computed in terms of the absolute root structure and the $\bbfQ$-rank of $G$. In particular, when $k$ is a number field of degree $d=[k:\bbfQ]$, and $G/\bbfQ=R\sb{k/\bbfQ}SL\sb n$, then $l(G/\bbfQ)\ge d(n-1)$. It is shown that this theorem implies the rank conjecture for all $K$-groups of an algebraic number field not equal to $\bbfQ$, in particular of the field of algebraic numbers $\overline\bbfQ$.
[W.W.J.Hulsbergen (Breda)]

MSC 2000:: *11F75 Cohomology of arithmetic groups
19D50 Computations of higher K-theory of rings
11R70 K-theory of global fields

Keywords: K-groups; semisimple algebraic group; real cohomology; rank filtration; rank conjecture

Cited in: Zbl 0841.19002

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