Non-expander Cayley graphs of simple groups

TL;DR

The paper constructs specific Cayley graphs of simple Lie groups with bounded degree whose isoperimetric numbers tend to zero, demonstrating these graphs are not expanders.

Contribution

It provides explicit examples of Cayley graphs of simple groups with bounded degree that are not expanders, challenging assumptions about expansion in such groups.

Findings

Cayley graphs of simple Lie groups with degree ≤ 10 have isoperimetric numbers approaching zero.

These graphs do not form a family of expanders despite increasing group rank.

The result applies to infinite sequences of simple groups of Lie type with growing rank.

Abstract

For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.