The super approximation property of ${\mathrm{SL}}_2(\mathbb{Z}/q\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z}/q\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z}/q\mathbb{Z})$

Abstract

Take $S\subset {\mathrm{SL}}_2(\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z})\times {\mathrm{SL}}_2(\mathbb{Z})$ be finite symmetric and assume $S$ generates a group $G$ which is Zariski-dense in ${\mathrm{SL}}_2\times {\mathrm{SL}}_2\times {\mathrm{SL}}_2(\mathbb{Z})$. This paper proves that the Cayley graphs $$\{\mathcal{C}ay(G(\mathrm{mod}q),S(\mathrm{mod}q)){\}}_{q\in {\mathbb{Z}}_{+}}$$ form a family of expanders.

arXiv admin note: substantial text overlap with arXiv:2308.09982 by other authors