Finiteness properties of certain arithmetic groups in the function field case. (English) Zbl 0819.20051

Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but in the case of global function fields the situation becomes more complicated. To attack this problem cohomological methods are often applied. A group \(\Gamma\) is called of type \(FP_ n\) if the \(Z\Gamma\)-module \(Z\) (with trivial \(\Gamma\)-module-structure, i.e. \(\gamma\cdot n = n\) for \(\gamma \in \Gamma\), \(n \in Z\)) admits a projective resolution which is finitely generated in dimensions \(\leq n\). The group \(\Gamma\) is of type \(FP_ 1\) if and only if \(\Gamma\) is finitely generated, and \(\Gamma\) is of type \(FP_ 2\) if it is finitely presented. Since the converse of the last assertion is not known one defines a group to be of type \(F_ n\) for \(n \geq 2\), if it is finitely presented and of type \(FP_ n\). The largest \(n\) such that \(\Gamma\) is of type \(F_ n\) is called the finiteness length of \(\Gamma \) and is denoted by \(\Phi(\Gamma)\). The main result of the paper under review is the following Theorem: \(\Phi(\text{SL}_ n(F_ q[t])) = n - 2\) if \(n \geq 2\) and \(q \geq 2^{n-2}\) (\(F_ q\) the finite field with \(q\) elements).
The proof of the theorem uses a careful study of the action of \(\Gamma\) on the Bruhat-Tits building \(X\) of \(\text{GL}_{n+1}\) and a general result of K. Brown on the cellular action of groups on contractible CW- complexes.


20G30 Linear algebraic groups over global fields and their integers
20J05 Homological methods in group theory
20F05 Generators, relations, and presentations of groups
20E42 Groups with a \(BN\)-pair; buildings
11R58 Arithmetic theory of algebraic function fields
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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