Commutator width in Chevalley groups. (Q2856433)

This is an expanded version of a conference talk given in Porto Cesareo in June 2011. It is an interesting account of recent results on the width of certain commutators in Chevalley groups with respect to certain generating sets of their elementary subgroups.NEWLINENEWLINE Let \(\Phi\) be a reduced irreducible root system and \(R\) be a commutative ring. Let \(G(\Phi,R)\) be a corresponding Chevalley group and let \(E(\Phi,R)\) be the subgroup generated by all its (elementary) root unipotents. It is well-known that, if \(\Phi\) has rank at least \(2\), then \(E(\Phi,R)\) is a \textit{normal} subgroup of \(G(\Phi,R)\) (for \textit{any} \(R\)). It follows therefore that every commutator \([x,y]\), where \(x\in G(\Phi,R)\) and \(y\in E(\Phi,R)\), is a product of root unipotents.NEWLINENEWLINE The main result discussed in this survey is the existence of an upper bound \(N(\Phi,R)\) for the number of unipotents required in such a product. The first versions required conditions on \(R\) similar to those introduced by Bass as part of classical algebraic K-theory. Later versions were able to remove these conditions. Remarkably there exists an upper bound \(N(\Phi)\) dependent \textit{only} on \(\Phi\). Many other extensions of this result are discussed. A list of related unsolved problems is also provided.NEWLINENEWLINE It is worth noting that the rank restriction on \(\Phi\) is necessary. For the case where \(G(\Phi,R)=\mathrm{SL}_2(\mathbb Z)\) (in which case \(R=\mathbb Z\), \(\mathrm{rk}(\Phi)=1\) and \(G(\Phi,R)=E(\Phi,R)\)), it is well-known that \([\mathrm{SL}_2(\mathbb Z),\mathrm{SL}_2(\mathbb Z)]\) does not have bounded width with respect to the root unipotents (which here are the elementary \(2\times 2\) matrices).