Expansion in $SL_d(O_K/I)$, $I$ square-free

TL;DR

This paper proves that Cayley graphs of certain special linear groups over residue rings form expander families when the ideals are square-free, extending known results to new cases involving number fields and specific dimensions.

Contribution

It establishes expansion properties for Cayley graphs of SL_d over residue rings for square-free ideals, covering new cases for d=2 over arbitrary number fields and d=3 over Q.

Findings

Cayley graphs form expander families for d=2 over any number field.

Expansion is proven for d=3 over Q with square-free ideals.

Results extend previous expansion theorems to broader algebraic settings.

Abstract

Let S be a fixed symmetric finite subset of SL_d(O_K) that generates a Zariski dense subgroup of SL_d(O_K) when we consider it as an algebraic group over Q by restriction of scalars. We prove that the Cayley graphs of SL_d(O_K/I) with respect to the projections of S is an expander family if I ranges over square-free ideals of O_K if d=2 and K is an arbitrary numberfield, or if d=3 and K=Q.