It is a classical result that the modulo map from SL2(ℤ) to SL2(ℤ/qℤ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL2(ℤ/qℤ) has a lift in SL2(ℤ) with coefficients of size at most qe? A simple pigeonhole principle shows that e is strictly greater than 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL2(ℤ). The proof relies on a density theorem of the Ramanujan conjecture for SL2(ℤ). In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands programme.
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