Geometric reductions of ABS equations on an n-cube to discrete Painlevé systems

In this paper, we show how to relate n-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the four-dimensional cube to the q-discrete third Painlevé equation, which is a dynamical system on a rational surface of type A5(1) with the extended affine Weyl group W-tilde ((A2+A1)(1)). We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A. (paper)