It is now well established that light trapping is an essential element of thin film solar cell design. Numerous light trapping geometries have already been applied to thin film cells, especially to silicon-based devices. Less attention has been paid to light trapping in GaAs thin film cells, mainly because light trapping is considered less attractive due to the material's direct bandgap and the fact that GaAs suffers from strong surface recombination, which particularly affects etched nanostructures. Here, we study light trapping structures that are implemented in a high-bandgap material on the back of the GaAs active layer, thereby not perturbing the integrity of the GaAs active layer. We study photonic crystal and quasi-random nanostructures both by simulation and by experiment and find that the photonic crystal structures are superior because they exhibit fewer but stronger resonances that are better matched to the narrow wavelength range where GaAs benefits from light trapping. In fact, we show that a 1500 nm thick cell with photonic crystals achieves the same short circuit current as an unpatterned 4000 nm thick cell. These findings are significant because they afford a sizeable reduction in active layer thickness, and therefore a reduction in expensive epitaxial growth time and cost, yet without compromising performance.