Optimal Lifting for the Projective Action of $S{L}_3(Z)$

Abstract

Let $\epsilon >0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $S{L}_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon }$, whose projection to $S{L}_3(F_q)$ 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 $S{L}_2(Z)$.

24 pages. Major revision. Added a proof of the simpler case of SL2, rewrote the proof of Theorem 1.4, added a theorem about optimal lifting for the action on flags

References in corpus (1)

• Cutoff on Hyperbolic Surfaces

Cited by in corpus (1)

• On the local $L^2$-Bound of the Eisenstein Series