Finiteness properties of solvable S-arithmetic groups: An example
A group \(\Gamma\) is of type \(FP_ n\) if the \(Z\Gamma\)-module Z admits a projective resolution which is finitely generated in dimensions \(\leq n\). For example, \(\Gamma\) is of type \(FP_ 1\) if and only if it is finitely generated, and \(\Gamma\) is of type \(FP_ 2\) if it is finitely presented. Stallings gave the first example of a group of type \(FP_ 2\) but not \(FP_ 3\), and Bieri extended this to a sequence of groups \(\Gamma_ n\) of type \(FP_{n-1}\) but not \(FP_ n\). Further examples of this type are due to Stuhler, whose groups are S-arithmetic groups \(SL_ 2(O_ S)\), where S is a finite set of primes in a function field and \(O_ S\) is the ring of S-integers. The authors exhibit similar sequences of S-arithmetic groups over number fields. The proof involves the Bruhat-Tits building; this, the authors remark, may well be the right approach for a systematic investigation. For \(n\geq 1\) and p a prime number, let \(\Gamma_ n\) be the group of upper triangular matrices g in \(GL_{n+1}(Z[1/p])\) with \(g(1,1)=g(n+1,n+1)=1\). Note that \(\Gamma_ n\) is a \(\{\) \(p\}\)-arithmetic subgroup of the solvable algebraic matrix group defined by the same equations as \(\Gamma_ n.\) Theorem A. \(\Gamma_ n\) is of type \(FP_{n-1}\) but not \(FP_ n\); for \(n\geq 3\) it is finitely presented. (The negative result (that \(\Gamma_ n\) is not of type \(FP_ n)\), the finite presentation for \(n\geq 3\), and the \(FP_{n-1}\) property for \(n\leq 4\) are known results. So the new result is that \(\Gamma_ n\) is of type \(FP_{n-1}\) for \(n\geq 5.)\) The positive part of Theorem A follows from: Theorem B. There is a simplicial action of \(\Gamma =\Gamma_ n\) on a simplicial complex X with the following properties: (a) X is (n-2)- connected. (b) The isotropy group of every simplex is finitely presented and of type \(FP_{\infty}\). (c) X is finite mod \(\Gamma\). The complex X is a subcomplex of the Bruhat-Tits building associated to the group \(GL_{n+1}\) and the p-adic valuation of \({\mathbb{Q}}\).
scientific article; zbMATH DE number 4000210
scientific article