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In this paper, we show how to relate $n$-dimensional cubes on which ABSequations hold to the symmetry groups of discrete Painlev\'e equations. We herefocus on the reduction from the 4-dimensional cube to the $q$-discrete thirdPainlev\'e equation, which is a dynamical system on a rational surface of type$A_5^{(1)}$ with the extended affine Weyl group $\widetilde{\mathcalW}\bigl((A_2+A_1)^{(1)}\bigr)$. We provide general theorems to show that thisreduction also extends to other discrete Painlev\'e equations at least of typeA.

Geometric Reductions of ABS equations on an $n$-cube to discrete Painlev\'e systems