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Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem. (English) Zbl 1290.20039
This is a landmark paper. The authors prove a conjecture that has been around for decades. The precise statement of the rank theorem is as follows. Let \(K\) be a global function field and let \(\mathcal G\) be a linear algebraic group defined over \(K\). Let \(S\) be a finite set of places of \(K\) and let \(\Gamma=\mathcal G(\mathfrak o_S)\) be the group of its \(\mathfrak o_S\)-points. This group is well defined only up to commensurability, but everything said hereafter depends only on the commensurability class of \(\Gamma\). A group \(G\) is called of type \(FP_m\) if it admits a classifying space with finite \(m\)-skeleton. The finiteness length \(\Phi(G)\) of \(G\) is the supremum \(\leq\infty\) of those \(m\) for which \(G\) is of type \(FP_m\). Let \(K_p\) be the completion of \(K\) at \(p\). The local rank \(d_p\) of \(\mathcal G\) at \(p\) is the rank of \(\mathcal G\) over \(K_p\). If \(\mathcal G\) is isotropic and almost simple, the group \(\mathcal G(K_p)\) acts on its associated Bruhat-Tits building \(X_p\) and the dimension of \(X_p\) is the local rank \(d_p\).
Here is the statement of the rank theorem. Let \(\mathcal G\) be a connected non-commutative absolutely almost simple \(K\)-isotropic \(K\)-group. Then the finiteness length of the \(S\)-arithmetic group \(\Gamma=\mathcal G(\mathfrak o_S)\) is \(d-1\), where \(d\) is the sum of the local ranks \(d_p\) for \(p\in S\).
This also determines the finiteness length of \(S\)-arithmetic subgroups of isotropic reductive groups. This result is in sharp contrast with the situation for \(S\)-arithmetic groups over number fields. Here all \(S\)-arithmetic subgroups of reductive groups have infinite finiteness length. This was proved by Raghunathan in 1968 for arithmetic groups, and in 1976 by Borel and Serre for the general case of \(S\)-arithmetic groups. Interest in the positive characteristic case started with the result of Nagao in 1959, who showed that \(\mathrm{SL}(2,\mathbb F_q[t])\) is not finitely generated. Behr showed in 1969 that \(\Gamma\) as in the rank theorem is finitely generated (\(=FP_1\)) if and only if \(d>1\) and in 1998 that \(\Gamma\) is finitely presented (\(=FP_2\)) if and only if \(d>2\). After Serre had noticed in the early seventies that the Bruhat-Tits building can be used to study the homology of \(S\)-arithmetic groups and Behr and Stuhler had shown by examples in the early eighties that the finiteness properties for isotropic groups do not depend on the global rank but on the sum of the local ranks, the “rank theorem” emerged as a conjecture, but was not declared a conjecture in print, at the time. Several special cases had been proved in the meantime.
The proof of the rank theorem given by the authors uses, like most of the proofs of the earlier results, the action of \(\Gamma\) on the product \(X\) of the Bruhat-Tits buildings \(X_p\) and reduction theory, here recast in terms of the metric structure of Euclidean buildings. Another ingredient of the proof is Ken Brown’s criterion which allows to compute finiteness properties of a group from a (nice) action of the group on an (appropriate) filtered CW-complex. Technically speaking, the authors use a combinatorial Morse function on \(X\). Their Morse function is \(\Gamma\)-invariant and has cocompact sublevel sets. The main point in the application of Ken Brown’s criterion is to show that descending links are highly connected. It is not too difficult to show that this is true generically. To prove that this holds for all descending links takes up a large part of the paper, and is achieved by perturbing the initial Morse function in several steps.

MSC:
20G30 Linear algebraic groups over global fields and their integers
51E24 Buildings and the geometry of diagrams
20F65 Geometric group theory
20E42 Groups with a \(BN\)-pair; buildings
20J05 Homological methods in group theory
11E57 Classical groups
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