Finite presentability of groups acting on locally finite twin buildings
Gates, Zachary, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Abramenko, Peter, Department of Mathematics, University of Virginia
Let G be a group acting strongly transitively on a locally finite twin building. The main examples of such groups are Kac-Moody groups over finite fields. In this case, G has an associated Weyl group W and accompanying Coxeter diagram. G is always finitely generated under these conditions, and it has been conjectured that if G has Coxeter diagram with at least one infinite label, then G is not finitely presented.
We prove for large classes of diagrams that G is not finitely presented. In doing so, we provide strong evidence that the conjecture is true by showing that G is not finitely presented if the diagram has exactly one infinite label and the rest is ``as spherical as possible". We also show that if G is rank 3 and has at least one infinite label in the diagram, then G is not finitely presented.
The main tool is a theorem of Gandini which states that a group acting on a 2-dimensional contractible space with certain properties is not finitely presented. We first introduce non-standard realizations of buildings since the twin building that G naturally acts on is, in general, of too high dimension. Under certain restrictions on the Coxeter diagram, we realize the twin building as a product of two trees on which G acts with the properties desired in order to apply Gandini's theorem.
PHD (Doctor of Philosophy)
Buildings, Twin buildings, Group theory, Kac-Moody groups, Finiteness properties
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