2009/07/08 by Rieuwert Blok, Blok, Rieuwert, Corneliu Hoffman +1
Mathematics · #20E42 #20G35 #51E24 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #math.AG #math.GR #msc:20E42 #msc:20G35 #msc:51E24
paper · pdf · doi:10.48550/arxiv.0907.1388
In this version it is emphasized that the main result is the classification of Curtis-Tits amalgams. The results themselves and their proofs are unchanged
arxiv created 2010/10/04 · arxiv updated 2010/10/05
A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that roughly speaking these groups are determined by their local structure. This result was later extended to Kac-Moody groups by P.~Abramenko and B.~Mühlherr. Their theorem states that a Kac-Moody group G is the universal completion of an amalgam of rank two (Levi) subgroups, as they are arranged inside G itself. Taking this result as a starting point, we define a Curtis-Tits structure over a given diagram to be an amalgam of groups such that the sub-amalgam corresponding to a two-vertex sub-diagram is the Curtis-Tits amalgam of some rank-2 group of Lie type. There is no a priori reference to an ambient group, nor to the existence of an associated (twin-) building. Indeed, there is no a priori guarantee that the amalgam will not collapse. We then classify these amalgams up to isomorphism. In the present paper we consider triangle-free simply-laced diagrams. Instead of using Goldschmidt's lemma, we introduce a new approach by applying Bass and Serre's theory of graphs of groups. The classification reveals a natural division into two main types: "orientable" and "non-orientable" Curtis-Tits structures. Our classification of orientable Curtis-Tits structures naturally fits with the classification of all locally split Kac-Moody groups using Moufang foundations. In particular, our classification yields a simple criterion for recognizing when Curtis-Tits structures give rise to Kac-Moody groups. The class of non-orientable Curtis-Tits structures is in some sense much larger. Many of these amalgams turn out to have non-trivial interesting completions inviting further study.