Groups generated by \(k\)-root subgroups
The paper is a successor of the author's paper [Invent. Math. 100, No. 1, 167-206 (1990; Zbl 0697.20018)]. The author studies groups \(G\) generated by so called \(k\)-root subgroups (\(k\) a commutative field). This is a set \(\Sigma\) of abelian groups with \(G= \langle \Sigma\rangle\) and \(\Sigma^ g=\Sigma\) for \(g\in G\) such that for each pair \(A,B\in\Sigma\) one of the following holds: (1) \([A,B]=1\), (2) \(\langle A,B\rangle\cong\text{(P)SL}(2,k)\) and \(A\) and \(B\) are full unipotent subgroups, (3) \(\langle A,B\rangle'\leq Z(\langle A,B\rangle)\) and \([a,B]= [b,A]= [A,B]\) is in \(\Sigma\) for all \(a\in A^ \#\) and \(b\in B^ \#\). Furthermore he assumes the maximality condition for unipotent \(\Sigma\)-subgroups. In the paper mentioned above the author treated the case that (3) never occurs. The conclusion is too lengthy to be stated here but roughly it says that there is a nilpotent normal subgroup \(R(G)\) such that \(G/R(G)\) is a quasisimple group, which then is a central extension of a well defined class of algebraic groups defined over certain extension fields of \(k\), which are described in a paper of \textit{J. Tits} [Proc. Symp. Pure Math. 9, 33-62 (1966; Zbl 0238.20052)] (misquoted as [25] in the paper under review). The proof of the theorem is geometric in nature. The author investigates three graphs on the set \(\Sigma\). This is the commuting graph, the graph whose edges are the pairs \((A,B)\) with \(\langle A,B\rangle \cong\text{(P)SL}(2,k)\), and a graph whose edges are pairs \((A,B)\) such that \(\Sigma\cap AB\) gives a partition of \(AB\). The latter is very important as the proof is done by induction on the number of generators from \(\Sigma\) needed to generate the group given by the neighbours of \(A\) in the last-mentioned graph. Investigation of these graphs together with a very detailed analysis of the groups generated by three members from \(\Sigma\) gives at the end that the group \(G\) has to act on some building, and then characterizations of these geometries yield the conclusion.
scientific article; zbMATH DE number 59788