Tropical carpetgrass (Axonopus compressus) can tolerate low mowing, even down to putting green height. Here it is invading a seashore paspalum (Paspalum vaginatum) nursery green.
Even though carpetgrass can grow at putting green heights, I don't recommend it for greens. In the locations where carpetgrass can grow, manilagrass (Zoysia matrella) can too, and I prefer manilagrass putting surfaces. Why? Manilagrass has a finer leaf blade and I expect the ball roll will be more predictable across manilagrass than across carpetgrass.
Carpetgrass on fairways -- I like that -- especially when mown at about 8 mm. And in the rough, under shade trees, tropical carpetgrass is the best choice in a tropical environment. These photos show more.