ON REPRESENTATION OF INTEGERS FROM THIN SUBGROUPS OF SLp2,Zq WITH PARABOLICS

Abstract. Let Λ ă SLp2,Zq be a finitely generated Fuchsian group of the second kind, and v,w be two primitive vectors in Z ́ 0. We consider the set S “ txvγ,wy : γ P Λu, where x ̈, ̈y is the standard inner product in R. Using some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, and Gamburd’s 5/6 spectral gap, we show that if Λ has parabolic elements, and the critical exponent δ of Λ exceeds 0.995037, then almost all admissible integers (i.e. integers passing all local obstructions) are actually in S, with a power savings on the size of the exceptional set (i.e. the admissible integers fail to appear in S). This supplements a result of BourgainKontorovich, which proves a similar statement for the case when Λ is free, finitely generated, has no parabolics and has critical exponent δ ą 0.999950.