Expander Graphs in Pure and Applied Mathematics

Abstract: Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics.

• The paper provides an exposition on expander graphs, detailing their definition, properties, intricate constructions, and diverse applications across mathematics.
• Constructions of expander graphs are non-trivial, utilizing deep mathematical tools from algebra, number theory, and representation theory like spectral gaps and Kazhdan's property (T).
• Expander graphs find widespread applications in computer science (e.g., networks, codes), pure mathematics (e.g., number theory, group theory), and geometry (e.g., manifolds).

Expander Graphs in Pure and Applied Mathematics

The paper "Expander Graphs in Pure and Applied Mathematics" by Alexander Lubotzky explores the multifaceted nature of expander graphs, their constructions, and diverse applications in both pure and applied mathematics. Expander graphs, defined as highly connected yet sparse graphs, serve as essential tools in various domains such as computer science, number theory, group theory, and geometry. This expository piece explores the intricate constructions of expander graphs, their properties, and significant contributions to theoretical and practical problems.

Definition and Properties of Expander Graphs

Expander graphs are defined on a set of vertices with specific conditions ensuring that they are well-connected and sparse. A graph $X$ is called an $\epsilon$-expander if, for every subset of vertices, certain conditions involving edge boundaries and vertex subsets are satisfied. The paper outlines the significance of eigenvalues, particularly the spectral gap, in determining the expansion properties of a graph. Exploiting these properties, expander graphs have emerged as crucial constructs in designing efficient networks and algorithms.

Constructions of Expander Graphs

The construction of expander graphs is non-trivial and has motivated deep mathematical inquiry. Random graphs often serve as models for expanders, but explicit constructions demand sophisticated methods. The paper highlights techniques involving Kazhdan's property (T), the Ramanujan conjecture, and other representation-theoretic tools. These constructions underscore the interplay between algebraic methods and combinatorial structures, which enables the realization of expander families suitable for various applications.

Applications in Computer Science

Expander graphs find significant applications in computer science, particularly in network construction, error-correcting codes, and derandomization techniques. For instance, they are used to develop low-density parity-check (LDPC) codes, which are vital in reliable data transmission over noisy channels. Additionally, the paper discusses their role in optimizing random algorithms, providing the theoretical underpinnings for algorithms like the product replacement algorithm, which efficiently generates random group elements.

Impact on Pure Mathematics

In pure mathematics, expander graphs contribute to an enhanced understanding of several branches, including number theory and group theory. The paper explores their relevance in unsolved conjectures about primes and orbits, demonstrating how these graphs apply the Brun sieve and affine sieve methods to study primes in specific orbits and linear groups. Moreover, they offer insights into group theoretical phenomena, such as characterizing generic group elements and exploring the growth rates of particular group classes.

Geometric Implications

Expander graphs also have implications in geometry, particularly in relation to hyperbolic manifolds. The paper examines how these graphs are involved in geometric group theory, affecting properties like the virtual Haken conjecture for 3-manifolds and notions of large 3-volumes in geometry. The connections between the Heegaard genus of manifolds, property $(\tau)$, and expander graphs establish an unexpected but significant relationship between algebraic properties and geometric configurations.

Future Directions and Open Questions

Looking ahead, the paper suggests several open questions and areas for future exploration, such as the challenge of extending the notion of expanders to higher dimensions and the implications of expanders in diverse fields like $C^*$-algebra and metric embeddings. Additionally, it poses questions regarding the uniform expansion of finite simple groups and the role of random generators in understanding the broader class of expanders.

In conclusion, the exposition by Lubotzky illuminates the expansive reach of expander graphs across mathematical disciplines, highlighting their profound influence and potential for facilitating future advances in theory and application. The insights gained from their study continue to inspire further research into both the foundational and applied dimensions of mathematics.