Asymptotic classes of finite Moufang polygons

In this paper we study the model theory of classes of finite Moufang polygons. We show that each family of finite Moufang polygons forms an ‘asymptotic class’. As a result, since every non-principal ultraproduct of an asymptotic class is ‘measurable’, and therefore supersimple of finite rank, we obtain examples of (infinite) supersimple Moufang polygons of finite rank. In a forthcoming paper, [8], we will show that all supersimple Moufang polygons of finite rank arise over supersimple fields and belong to exactly those families which also have finite members. This body of work will give a description of groups with supersimple finite rank theory which have a definable spherical Moufang BN-pair of rank at least two.