Representations of the Gupta-sidki Group

If p is an odd prime, then the Gupta-Sidki group Gp is an infinite 2-generated p-group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree p. In this note, we make two observations concerning the irreducible representations of the group algebra K[Gp] with K an algebraically closed field. First, when charK 6= p, we obtain a lower bound for the number of irreducible representations of any finite degree n. Second, when charK = p, we show that if K[Gp] has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup Hp of Gp has a normal subgroup of finite index isomorphic to the direct product of p copies of Hp. 1. Representations of finite degree If p is an odd prime, then we let Gp denote the Gupta-Sidki group described in [GS]. Specifically, Gp is an infinite 2-generated p-group defined in a recursive manner as a subgroup of the automorphism group of a regular tree of degree p. Consequently, Gp is residually finite. In fact, Gp is just infinite [S1] in the sense that every nontrivial normal subgroup of Gp has finite index. For our purposes, a key property of this group is that if Hp is its commutator subgroup, then Hp has a normal subgroup of finite index isomorphic to the direct product of p copies of itself (see [S1]). IfG is any finitely generated group and if all finite-dimensional K-representations of G are completely reducible, then a theorem of Weil (see [F]) implies that G has only finitely many K-representations of any given finite degree n. In particular, this applies to the Gupta-Sidki group Gp provided charK 6= p, and motivates the following Definition. Let K be a fixed algebraically closed field and, for any group G, let fG(n) be the number of irreducible representations of K[G]of degree n. Furthermore, let FG(n) = ∑n i=1 fG(i) be the number of irreducible representations of K[G] of degree ≤ n. Thus, fG(n) and FG(n) are either nonnegative integers or ∞. Of course, irreducible representations of K[G] correspond in a natural manner to irreducible K[G]-modules. We first consider the behavior of the functions fG and FG with respect to normal subgroups of finite index. Received by the editors November 8, 1994. 1991 Mathematics Subject Classification. Primary 20C07; Secondary 16S34, 20E08, 20F50. The first author’s research supported in part by NSF Grant DMS-9224662. c ©1996 American Mathematical Society