On the expansion rate of Margulis expanders

Abstract

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,{A formula is presented}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. © 2005 Elsevier Inc. All rights reserved.