Thin Groups of Prime-Power Order and Thin Lie Algebras: An Addendum
Abstract
By A. CARANTIt [Received 2 May 1997; in revised form 16 September 1997] Introduction IN [2] we have shown that certain modular graded Lie algebras called thin Lie algebras T are uniquely determined by their factors of low dimension. This was achieved by constructing certain graded Lie algebras L which are finitely presented, and by showing that their central quotients L / Z (L) are isomorphic to T. The question left open in [2] was how big Z(L) actually was. As we will see below, this is equivalent to asking whether T itself is finitely presented or not. This was not critical for our purposes in [2], so we were content with giving an upper bound for Z(L), and by providing computational evidence for the fact that Z(L) ::j:. {O}. In this short note we determine Z(L) precisely: it turns out to be infinite-dimensional. In [2] the Lie algebras T were determined by grading special linear Lie algebras of dimension one and two over the p-adic integers, and this approach was used in [6] to construct the corresponding groups. The idea here is first to rewrite the algebras of [2] in terms of loop algebras of special linear