Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2(R)

An, Jinpeng ; Ghosh, Anish ; Guan, Lifan ; Ly, Tue (2019) Bounded orbits of diagonalizable flows on finite volume quotients of products of SL2(R) Advances in Mathematics, 354 . p. 106743. ISSN 0001-8708

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Official URL: http://doi.org/10.1016/j.aim.2019.106743

Related URL: http://dx.doi.org/10.1016/j.aim.2019.106743

Abstract

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let G := SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) and \Gamma be a lattice in G. We show that the set of points on G/\Gamma whose forward orbits under a one parameter Ad-semisimple subsemigroup of G are bounded, form a hyperplane absolute winning set.

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