On the expansion rate of Margulis expanders

In this note we determine exactly the expansion rate of an infinite 4-regular expander graph which is a variant of an expander due to Margulis. The vertex set of this graph consists of all points in the plane. The point (x, y) is adjacent to the points S(x, y), S(x, y), T (x, y), T(x, y) where S(x, y) = (x, x+y) and T (x, y) = (x+y, y). We show that the expansion rate of this 4-regular graph is 2. The main technical result asserts that for any compact planar set A of finite positive measure, |S(A) ∪ S(A) ∪ T (A) ∪ T(A) ∪A| |A| ≥ 2, where |B| is the Lebesgue measure of B. The proof is completely elementary and is based on symmetrization a classical method in the area of isoperimetric problems. We also use symmetrization to prove a similar result for a directed version of the same graph.