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Super Approximation for subgroups of $\text{SL}_d(\mathbb Z)$

Algebraic Geometry and Number Theory

Speaker: Xin Zhang, University of Hong Kong
Related Webpage: https://sites.google.com/view/mathzhangxin
Location: 2112 MSB
Start time: Thu, Feb 6 2025, 1:10PM

It is a discovery of Margulis in 1970s that congruence quotients of $\text{SL}_2(\mathbb Z)$ can be used to construct expanders, which are certain sparse but highly connected graphs and are ideal models for network building.  The Super Approximation Conjecture of Salehi-Golsefidy and Varju gives a precise prediction on which more general subgroups of $\text{SL}_d(\mathbb Z)$ have this property. In this talk, I will survey the history of this conjecture, and describe a recent progress by Tang Jincheng and myself that all Zariski dense subgroups of $\text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbb Z^2$ have this property. This progress relies on the development of a key tool in arithmetic combinatorics conjectured by Salehi-Golsefidy.