Consider the group scheme

$$ R \mapsto G(R) := R \rtimes R^* = \left( \begin{array}{cc} 1 & R \\ 0 & R^* \end{array} \right) = \left( \begin{array}{cc} 1 & *\\ 0 & * \end{array} \right) \cap \mathrm{GL}_2(R) $$

where $R$ is an arbitrary commutative ring with $1 \neq 0$ and a unit $x \in R^*$ acts on $R$ by multiplication.

We will study the finiteness properties of subgroups of $G(\Oka_S)$ where $\Oka_S$ is an $S$-arithmetic subring of a global function field. The subgroups we are interested in are of the form $\Oka_S \rtimes Q$ where $Q$ is a subgroup of $\Oka_S^*$. The finiteness properties of these metabelian groups can be expressed in terms of the $\Sigma$-invariant due to R. Bieri and R. Strebel.

{\sc Theorem A.} {\it Let $S$ be a finite set of places of a global function field (regarded as normalized discrete valuations) and $\Oka_S$ the corresponding $S$-arithmetic ring. Let $Q$ be a subgroup of $\Oka_S^*$. Then $Q$ is finitely generated and for all integers $n \geq 1$ the following are equivalent:

\begin{enumerate} \item[\rm (1)] $\Oka_S \rtimes Q$ is of type FP$_n$;

\item[\rm (2)] $\Oka_S$ is $n$-tame as a $\ZZZ Q$-module;

\item[\rm (3)] each $p \in S$ restricts to a non-trivial homomorphism $p|_{Q} : Q \rightarrow \RRR$ and the set $\{ [p|_Q] \mid p \in S \} \subseteq \SSS(Q)$ is $n$-tame. \end{enumerate}

If these conditions hold for at least one $n \geq 1$ then the identity

$$ \Sigma_{\Oka_S}^c(Q,\ZZZ) = \{ [p|_Q] \mid p \in S \} $$

holds.} {\sc Theorem B.} {\it Let $r$ denote the rank of $Q$. Then the following hold: \begin{enumerate} \item[\rm (1)] the group $\Oka_S \rtimes Q$ is not of type FP$_{r+1}$;

\item[\rm (2)] if $Q$ has maximum rank $r=|S|-1$ then the group $\Oka_S \rtimes Q$ is of type FP$_r$.

\end{enumerate} In particular, $G(\Oka_S) = \Oka_S \rtimes \Oka_S^*$ is of type FP$_{|S|-1}$ but not of type FP$_{|S|}$.}

1991 Mathematics Subject Classification: 20E08, 20F16, 20G30, 52A20.