In this talk we discuss the main results of my PhD thesis. We begin by giving some background on Moufang polygons. This is followed by a short introduction of the basic model theoretic notions related to the thesis, such as asymptotic classes of finite structures, measurable structures, (superstable) supersimple theories and (finite Morley rank) S_1 rank. We also mention the relation between Moufang polygons and the associated little projective groups.
Moufang polygons have been classified by Tits and Weiss, and a complete list is given in their book `Moufang polygons'.
This work is inspired by a paper of Kramer, Tent and van Maldeghem called "Simple groups of finite Morley rank and Tits buildings". The authors work in a superstable context. They show that Moufang polygons of finite Morley rank are exactly Pappian polygons, i.e., projective planes, symplectic quadrangles and split Cayley hexagons, provided that they arise over algebraically closed fields.
We work under the weaker assumption of supersimplicity. Therefore, we expect more examples. Indeed, apart from those already occuring in the finite Morley rank case, there are four further examples, up to duality, of supersimple Moufang polygons; namely, Hermitian quadrangles in projective dimension 3 and 4, the twisted triality hexagon and the (perfect) Ree-Tits octagon, provided that the underlying field (or `difference' field in the last case) is supersimple.
As a result, we obtain the nice characterization that supersimple Moufang polygons are exactly those Moufang polygons belonging to families which also arise over finite fields.
Examples of supersimple Moufang polygons are constructed via asymptoticity
arguments: every class C of finite Moufang polygons forms an asymptotic class, and every non-principal ultraproduct of C gives rise to a measurable structure, thus supersimple (of finite S_1 rank). For the remaining cases one can proceed as follows: let \Gamma be any Moufang polygon belonging to a family which does not arise over finite fields, and call K its underlying field; then K is
(first-order) definable in \Gamma, and by applying some model theoretic facts this definability is inconsistent with supersimplicity".