On the Topological Generation of Exceptional Groups by Unipotent Elements

Transformation Groups https://doi.org/10.1007/s00031-023-09798-0 Transformation Groups On the Topological Generation of Exceptional Groups by Unipotent Elements Timothy C. Burness1 Received: 21 June 2022 / Accepted: 1 March 2023 © The Author(s) 2023 Abstract Let G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p  0 which is not algebraic over a finite field. Let C1 , . . . , Ct be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if t  5 (or t  4 if G = G 2 ), then there exist elements xi ∈ Ci such that x1 , . . . , xt  is Zariski dense in G. Moreover, this bound on t is best possible. Here we establish a more refined version of this result in the special case where p > 0 and the Ci are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes C1 , . . . , Ct for t  2 such that x1 , . . . , xt  is Zariski dense for some xi ∈ Ci . 1 Introduction Let G be a simple algebraic group over an algebraically closed field k of characteristic p  0 and let C1 , . . . , Ct be non-central conjugacy classes in G, where t  2. Consider the irreducible subvariety X = C1 × · · · × Ct of the Cartesian product G t . Given a tuple x = (x1 , . . . , xt ) ∈ X , let G(x) denote the Zariski closure of the subgroup x1 , . . . , xt  and set  = {x ∈ X : G(x) = G}. In this setting, a basic problem is to determine whether or not  is empty. Note that if k is algebraic over a finite field then G is locally finite and thus  is always empty, so this problem is only interesting when k is not algebraic over a finite field, which is a hypothesis we adopt throughout the paper. This problem has been the subject of several recent papers and some general results have been established. In characteristic zero, for example, a theorem of Guralnick [13] shows that  is always an open subset of X . In the general setting, [3, Theorem 2] states that  is non-empty if and only if it is dense in X , while [4, Theorem 1] reveals B Timothy C. Burness 1 T.C. Burness, School of Mathematics, University of Bristol, Bristol BS8 1UG, UK Timothy C. Burness that  is non-empty if and only if it is generic, which means that it contains the complement of a countable union of proper closed subvarieties of X . If G is a simple algebraic group of exceptional type, then [3, Theorem 7] states that  is non-empty if t  5, and the same conclusion holds for t  4 when G = G 2 . Moreover, this is best possible since there are examples where t = 4 (t = 3 for G = G 2 ) and  is empty (see [3, Theorem 3.22]). In this paper, we extend the earlier work in [3] by studying the special case where G is an exceptional type group in positive characteristic p and each Ci is a conjugacy class of unipotent elements of order p (note that every nontrivial unipotent element has order p if p  h, where h is the Coxeter number of G). In this situation, our aim is to classify the varieties X = C1 × · · · × Ct where t  2 and  is empty. This is in a similar spirit to the main theorem of [4], which considers the analogous problem for symplectic and orthogonal groups when the Ci comprise elements of prime order modulo the centre of the group. In turn, this extends earlier work of Gerhardt [11] on linear groups. Notice that all of these results are independent of the isogeny class of G since the centre of G is contained in the Frattini subgroup and thus a subgroup H is dense in G if and only if H Z /Z is dense in G/Z , where Z is any central subgroup of G. We will typically work with the simply connected form of the group. Our main result is the following (Tables 12 and 13 are presented at the end of the paper in Section 6). Note that any two involutions generate a dihedral group, which explains why we assume p  3 when t = 2. Theorem 1 Let G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p > 0 that is not algebraic over a finite field. Set X = C1 × · · · × Ct , where t  2 and each Ci is a conjugacy class of elements of order p in G. Assume p  3 if t = 2. Then  is empty if and only if X is one of the cases recorded in Tables 12 and 13. Remark 1 Some remarks on the statement of Theorem 1 are in order. (a) We adopt the notation for unipotent classes from [21] (see Tables 22.1.1−22.1.5 in [21]) and it is worth noting that this sometimes differs from the notation used by other authors. For example, the unipotent class in E 7 labelled (A31 )(1) in [21] is denoted (3A1 ) in [18, 25]. Similarly, the class E 6 (a3 ) in [18, 21] is labelled A5 + A1 in [25]. (b) By inspecting the relevant tables in [18], it is easy to read off the required condition on p to ensure that each Ci contains elements of order p. To do this, we consider the action of G on a suitable kG-module V and we inspect the Jordan form of a representative y ∈ Ci in this representation, noting that y has order p if and only if every Jordan block has size at most p. For example, if G = E 8 and y ∈ G is contained in the class A2 , then [18, Table 9] states that the Jordan form of y on the adjoint module for G is as follows ⎧ 2 54 78 ⎨ (J4 , J3 , J1 ) p = 2 p=3 (J 57 , J 77 ) ⎩ 3 551 78 (J5 , J3 , J1 ) p  5 On the Topological Generation of Exceptional Groups by Unipotent Elements Table 1 The varieties Y = C1 × · · · × Ct in Corollary 2 G (C1 , . . . , Ct ) G2 (A1 , A1 , A1 ), (A1 , G 2 (a1 )) F4 (A1 , A1 , A1 , A1 ), (A1 , A1 , A2 ), (A1 , Ã1 , Ã1 ), (A1 , B3 ), ( Ã1 , Ã2 ), ( Ã1 , B2 ), (A2 , A2 ) E6 (A1 , A1 , A1 , A1 ), (A1 , A1 , A2 ), (A1 , A21 , A21 ), (A1 , D4 ), (A1 , A4 ), (A21 , A3 ), (A21 , A22 ), (A2 , A2 ) E7 (A1 , A1 , A1 , A1 ), (A1 , A1 , A2 A1 ), (A1 , (A31 )(1) , (A31 )(1) ), (A1 , (A5 )(1) ), (A1 , D5 (a1 )), ((A31 )(1) , (A3 A1 )(1) ), (A2 , A2 A1 ) E8 (A1 , A1 , A1 , A21 ), (A1 , A1 , A3 ), (A1 , A21 , A2 ), (A1 , D5 ), (A1 , D4 A2 ), (A21 , D4 ), (A2 , A3 ) where Ji denotes a standard unipotent Jordan block of size i. Therefore, the elements in this class have order p if and only if p  3 (they have order 4 when p = 2). (c) We have chosen to record the cases X = C1 × · · · × Ct with  empty over two tables, rather than one. This is essentially an artefact of our proof and it will be convenient to make a distinction between the cases in Tables 12 and 13 (for example, see Theorem 3.1 below). (d) To avoid unnecessary repetition, the tuples in Tables 12 and 13 are listed up to reordering, and also up to graph automorphisms when (G, p) = (G 2 , 3) or (F4 , 2). For example, if (G, p) = (G 2 , 3) and τ is a graph automorphism of G, then τ interchanges the classes of long and short root elements (denoted by A1 and Ã1 ), whence  is also empty when t = 3 and (C1 , C2 , C3 ) = ( Ã1 , Ã1 , Ã1 ). The Zariski closure of a unipotent conjugacy class is a union of unipotent classes and this leads naturally to a partial order on the set of un (...truncated)