Abstract

An expander is an infinite family of finite graphs, with a growing number of vertices, that are low vertex degree yet highly connected. Expanders are ubiquitous in mathematics and computer science. In this talk, we focus on expanders with girth tending to infinity. First, we briefly indicate their importance for recent results in group theory, metric geometry and operator K-theory. Then we discuss our new explicit construction of finite 4-regular graphs as in the title. For each dimension $n \geq 2$, our graphs are suitable Cayley graphs of $SL_n(\mathbb{F}_p)$ as prime $p \rightarrow \infty$. These are the first explicit examples in all dimensions $n \geq 2$ (all prior examples were in $ n=2$). Together with Margulis' and Lubotzky-Phillips-Sarnak's classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders. This is a joint work with Arindam Biswas.