Hyperplane absolute winning property of bounded orbits under diagonalizable flows on ${\mathrm{SL}}_3(\mathbb{C})/{\mathrm{SL}}_3({\mathcal{O}}_{\mathbb{K}})$

Abstract

We extend the work of An, Guan and Kleinbock on bounded orbits of diagonalizable flows on ${\mathrm{SL}}_3(\mathbb{R})/{\mathrm{SL}}_3(\mathbb{Z})$ to ${\mathrm{SL}}_3(\mathbb{C})/{\mathrm{SL}}_3({\mathcal{O}}_{\mathbb{K}})$, where $\mathbb{K}$ is an imaginary quadratic field. To achieve this, we first prove a complex analogue of Minkowski's Linear Forms Theorem. We then set up an appropriate Schmidt game in ${\mathbb{C}}^3$ such that bounded orbits correspond to a hyperplane-absolute-winning set consisting of certain vectors in ${\mathbb{C}}^3$ relative to an approximation by imaginary quadratic rationals in $\mathbb{K}$.

15 pages. Split previous Section 2 into two sections. Updated proofs