Genus, thickness and crossing number of graphs encoding the generating properties of finite groups

Assume that G is a finite group and let a and b be non-negative integers. We define an undirected graph Γa,b(G) whose vertices correspond to the elements of Ga∪Gb and in which two tuples (x1,…,xa) and (y1,…,yb) are adjacent if and only if 〈x1,…,xa,y1,…,yb〉=G. Our aim is to estimate the genus, the thickness and the crossing number of the graph Γa,b(G) when a and b are positive integers, giving explicit lower bounds on these invariants in terms of |G|.