Representations of the Gupta-Sidki group

If $p$ is an odd prime, then the Gupta-Sidki group ${\mathcal G}_p$ 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 $K[{\mathcal G}_p ]$ with $K$ an algebraically closed field. First, when $\operatorname {char} K\neq p$, we obtain a lower bound for the number of irreducible representations of any finite degree $n$. Second, when $\operatorname {char} K=p$, we show that if $K[{\mathcal G}_p ]$ 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 ${\mathcal H}_p$ of ${\mathcal G}_p$ has a normal subgroup of finite index isomorphic to the direct product of $p$ copies of ${\mathcal H}_p$.