Finite and Algorithmic Model Theory

146 Dugald Macpherson and Charles Steinhornwhich generate Sp 2m (q). Since the symplectic group is a product of a boundednumber, dependent only on m, of these root subgroups, it is itself uniformlyinterpretable in F q , and hence so is PSp 2m (q). In the other direction, to constructthe field F q inside PSL n (q), the additive structure is given by a root group. Themultiplicative structure of the field arises from a torus – which is conjugate tothe image in PSL n (q) of an appropriate diagonal subgroup of SL n (q) – actingon the root group. See also [59] and [42].For the families of Suzuki and Ree twisted simple groups, the situation israther more complicated. The construction involves an automorphism of theDynkin diagram which does not preserve lengths of roots. As a result, thesegroups are uniformly parameter bi-interpretable not with pure fields, but withdifference fields. The class of Suzuki groups 2 B 2 (2 2k+1 ) is uniformly parameterbi-interpretable with the clas C (1,2,2) , as is the class of Ree groups 2 F 4 (2 2k+1 ).The Ree groups 2 G 2 (3 2k+1 ) are uniformly parameter bi-interpretable with themembers of C (1,2,3) . In these cases we apply Example 4.3.2 above.Example 4.3.4 The families of simple groups of Lie type all arise as automorphismgroups of Tits buildings. The building blocks for these are the so-called‘rank 2 residues’, which are generalized polygons. Here, a generalized polygonis an incidence structure of points and lines such that the associated bipartiteincidence graph – which has the points and lines as vertices with incidencefor adjacency – has diameter n and girth 2n. A generalized n-gon is said tobe thin if it is an ordinary n-gon, and is thick if every point (respectively line)is incident with at least three lines (respectively points). A thick generalized3-gon is just a projective plane. The generalized polygons involved in finite simplegroups satisfy an additional symmetry condition, the ‘Moufang’ property.Moufang generalized polygons have been classified by Tits and Weiss [61].In particular, there are seven families of finite Moufang generalized polygons,each such polygon associated with its corresponding ‘little projective group.’Dello Stritto [19] shows that each of these seven families forms an asymptoticclass by proving that the polygons are uniformly parameter bi-interpretablewith their corresponding little projective groups, as each corresponding classof groups forms an asymptotic class, by Example 4.3.3 above.Example 4.3.5 By [48, Theorem 3.14], the collection of all finite cyclicgroups is a 1-dimensional asymptotic class. This is hardly surprising, as themultiplicative groups of finite fields are cyclic. In general, the result followsfrom Szmielew’s Theorem (see for example [31, Theorem A.2.2]), which saysthat in every abelian group every formula ϕ(x,ȳ) is equivalent to a booleancombination of formulas of the form p m |t(x,ȳ) ort(x,ȳ) = 0, where p is