Words and characters in finite p-groups

Given a group word w in k variables, a finite group G and g∈G, we consider the number Nw,G(g) of k-tuples (g1,…,gk) of elements of G such that w(g1,…,gk)=g. In this work we study the functions Nw,G for the class of nilpotent groups of nilpotency class 2. We show that, for the groups in this class, Nw,G(1)≥|G|k−1, an inequality that can be improved to Nw,G(1)≥|G|k/|Gw| (Gw is the set of values taken by w on G) if G has odd order. This last result is explained by the fact that the functions Nw,G are characters of G in this case. For groups of even order, all that can be said is that Nw,G is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when Nxn,G is a character if G is a 2-group of nilpotency class 2. Finally we also address the (much harder) problem of studying if Nw,G(g)≥|G|k−1 for g∈Gw, proving that this is the case for the free p-groups of nilpotency class 2 and exponent p.