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Optimal lifting for the projective action of $\operatorname{SL}_3(\mathbb{Z})$

Amitay Kamber and Hagai Lavner

Vol. 17 (2023), No. 3, 749–774
Abstract

Let 𝜖 > 0 and let q be a prime. We prove that with high probability, given x, y in the projective plane over 𝔽q, there exists γ SL 3(), with coordinates bounded by q13+𝜖, whose projection to SL 3(𝔽q) sends x to y. The exponent 1 3 is optimal and the result is a high rank generalization of Sarnak’s optimal strong approximation theorem for SL 2().

Keywords
optimal lifting, congruence subgroups
Mathematical Subject Classification
Primary: 11F06
Milestones
Received: 9 August 2021
Revised: 14 March 2022
Accepted: 10 May 2022
Published: 12 April 2023
Authors
Amitay Kamber
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom
Hagai Lavner
Einstein Institute of Mathematics
The Hebrew University of Jerusalem
Jerusalem
Israel

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