On expanders from the action of GL(2,Z)

Consider the undirected graph Gn = (Vn, En) where Vn = (Z/nZ) and En contains an edge from (x, y) to (x+ 1, y), (x, y + 1), (x+ y, y), and (x, y + x) for every (x, y) ∈ Vn. Gabber and Galil, following Margulis, gave an elementary proof that {Gn} forms an expander family. In this expository note, we present a somewhat simpler proof of this fact, and demonstrate its utility by isolating a key property of the linear transformations (x, y) 7→ (x+y, x), (x, y+x) that yields expansion. As an example, take any invertible, integral matrix S ∈ GL2(Z) and let Gn = (Vn, E n ) where E n contains, for every (x, y) ∈ Vn, an edge from (x, y) to (x + 1, y), (x, y + 1), S(x, y), and S>(x, y), and S> denotes the transpose of S. Then {Gn} forms an expander family if and only if the infinite graph G = ( Z \ {0}, { {z, Sz}, {z, S>z} : z ∈ Z \ {0} }) has positive Cheeger constant. This latter property turns out to be elementary to analyze: For any S = ( a b c d ) ∈ GL2(Z), the graph G has positive Cheeger constant if and only if (a+d)(b− c) 6= 0. The case S = ( 1 1 0 1 ) recovers the Margulis-Gabber-Galil graphs. We also present some other generalizations.