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 - 1 ( x , y ) , T ( x , y ) , T - 1 ( 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 - 1 ( A ) ∪ T ( A ) ∪ T - 1 ( A ) ∪ A | | A | ≥ 2 , where | B | is the Lebesgue measure of B. oof 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.