On expanders from the action of GL(2,Z)
James R. Lee
TL;DR
This work provides an elementary, unified framework linking Gabber-Galil expanders to a family of GL$_2(\mathbb{Z})$-generated graphs. By relating finite $G_n^{S,T}$ to an infinite graph $G^{S^{\top},T^{\top}}$ via the Cheeger constant and employing a torus-based Fourier analysis, it yields explicit expansion criteria: $\{G_n^{S,S^{\top}}\}$ expands precisely when $S \neq S^{\top}$ and $\operatorname{tr}(S)\neq 0$, and $\{G_n^{S,RSR}\}$ expands when $(a+d)(b+c)\neq 0$ for $S=(\begin{smallmatrix}a&b\\c&d\end{smallmatrix})$. The approach also identifies cases where expansion fails (e.g., $T=S^{-1}$ or $S^4=I$) and provides a versatile toolkit for analyzing a broad class of generalized expanders through elementary, combinatorial and harmonic-analytic methods on lattices and tori.
Abstract
Consider the undirected graph $G_n=(V_n, E_n)$ where $V_n = (Z/nZ)^2$ and $E_n$ contains an edge from $(x,y)$ to $(x+1,y)$, $(x,y+1)$, $(x+y,y)$, and $(x,y+x)$ for every $(x,y) \in V_n$. Gabber and Galil, following Margulis, gave an elementary proof that ${G_n}$ forms an expander family. In this note, we present a somewhat simpler proof of this fact, and demonstrate its utility by isolating a key property of the linear transformations $(x,y) -> (x+y,x), (x,y+x)$ that yields expansion. As an example, consider any invertible, integral matrix $S \in GL_2(Z)$ and let $G^S_n = (V_n, E^S_n)$ where $E^S_n$ contains, for every $(x,y) \in V_n$, an edge from $(x,y)$ to $(x+1,y)$, $(x,y+1)$, $S(x,y)$, and $S^T(x,y)$, where $S^T$ denotes the transpose of $S$. Then {G_n^S} forms an expander family if and only if a related infinite graph has positive Cheeger constant. This latter property turns out to be elementary to analyze and can be used to show that {G_n^S} are expanders precisely when the trace of S is non-zero and S is not equal to its transpose. We also present some other generalizations.