Simplicial complexes associated to character degrees of solvable groups

Graphs associated to the set of irreducible character degrees of a finite group $G$ have been extensively studied as a way of understanding structure of the underlying group. Another approach, proposed by Isaacs, is to study associated simplicial complexes, namely, the common divisor simplicial complex ${𝒢}(G)$ and the prime divisor simplicial complex ${𝒟}(G)$. These complexes can be associated to any set of positive integers and this paper shows that they are homotopy equivalent. Further, considering these complexes associated to the set of irreducible character degrees, we give a bound on the rank of the fundamental group.