The Farrell-Jones conjecture for \(S\)-arithmetic groups. (English) Zbl 1365.18010

The main purpose of the paper under review is the proof of the Farrell-Jones Isomorphism Conjecture, with finite wreath products, to the groups \(\mathrm{GL}_n(\mathbb{Q})\) and \(\mathrm{GL}_n(F(t))\), where \(F\) is a finite filed. In particular, it is proved that the conjecture holds for \(S\)-arithmetic groups. The main tool is a technical result result in [A. Bartels et al., Publ. Math., Inst. Hautes Étud. Sci. 119, 97–125 (2014; Zbl 1300.19001)]. To describe the result, we start with a group \(G\) that acts properly, and isometrically on a CAT(0) space \(X\), of covering dimension \(\leq N\). Also, there is a collection \(\mathcal{W}\) of open subsets of \(X\), of dimension \(\leq N\), which is \(G\)-invariant and the \(G\) action sends each element to itself or to a disjoint element of \(\mathcal{W}\). The last property that should be satisfied is that the group \(G\) acts cocompactly away from \(\beta\)-neighborhoods of elements of \(\mathcal{W}\), for each \(\beta > 0\). Under these conditions, the flow space \(FS(X)\) admits long \(\mathcal{F}\)-covers at infinity for the family of subgroups that consists of the virtually cyclic subgroups of \(G\) and the isotropy groups of the elements of \(\mathcal{W}\). That means that the \(K\)-theoretic Farrell-Jones Conjecture, with finite wreath products, holds for the family \(\mathcal{F}\) and the \(L\)-theoretic Farrell-Jones Conjecture holds for the family \(\mathcal{F}_2\), consisting of all subgroups that contains a subgroup in \(\mathcal{F}\) of index 2.
The author gives the construction for the space and the collection of open subsets in the above result in the cases of \(\mathrm{GL}_n(\mathbb{Z})\) and \(\mathrm{GL}_n(Z[T^{-1}])\), where \(Z = F[t]\), \(F\) a finite field and \(T\) a finite set of primes in \(Z\). In the first case, he starts with a free \(\mathbb{Z}\)-module of rank \(n\). Let \(\tilde{X}(V)\) be the space of inner products on \(\mathbb{R}{\otimes}_{\mathbb{Z}}V\) and \(X(V)\) is the quotient of \(\tilde{X}(V)\) under the group action of the multiplicative group \(\mathbb{R}^{> 0}\). In the case of \(\mathrm{GL}_n(Z[T^{-1}])\) the space \(X\) is the building \(X(V)\) accosisted to a valuation of the fraction field \(Q\) of \(Z\) and \(V\) is free \(F[t]\)-module. The group action comes from the fact that \(\text{aut}_{F[t]}(V) = \mathrm{GL}_n(F[t])\), which acts naturally on the building. The main technical point is the definition of the logarithmic volume in the above cases that is needed in the proof of the last part in the above statement. In the second case, the proof is completed in the reduction of the class of subgroups. The first reduction is that the \(K\)- and \(L\)-theoretic Farrell-Jones Conjecture with finite wreath products for the family consisting of virtually cyclic subroups and the stabilizers of non-trivial direct summands of \(Z[T^{-1}]^n\). The second reduction is to the family of virtually solvable subgroups. The third step is the proof of the actual conjecture, namely the reduction to class of virtually cyclic subgroups.


18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
19A31 \(K_0\) of group rings and orders
19B28 \(K_1\) of group rings and orders
19G24 \(L\)-theory of group rings


Zbl 1300.19001
Full Text: DOI arXiv


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