Spectral gap and origami expanders

TL;DR

This paper constructs new measure-preserving affine actions with spectral gap on surfaces of any genus, leading to novel expanders distinct from classical ones, and demonstrates coarse distinctions among various expanders.

Contribution

It introduces the first measure-preserving affine actions with spectral gap on arbitrary genus surfaces and constructs new expanders that differ from classical Laplacian-based expanders.

Findings

Constructed measure-preserving affine actions with spectral gap on surfaces of genus > 1.

Developed geometric representatives of multi-twists on origami surfaces.

Produced new expanders coarsely distinct from classical Laplacian-based expanders.

Abstract

We construct the first measure-preserving affine actions with spectral gap on surfaces of arbitrary genus $g > 1$. We achieve this by finding geometric representatives of multi-twists on origami surfaces. As a major application, we construct new expanders that are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. Our methods also show that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander.