What I've read the process to get to 1" cut height from the more typical 3"+ is the following:
- Dethatch to remove thatch and thin out the crown. Do this with a power rake, verticutter, seed slitter, or a dethatcher.
- Cut lower than what your target height is. This will scalp to a certain extent but will "reset" the crown height. The lawn will turn brown but in a couple weeks it'll green back up. (Do this when its not crazy hot out, i.e. spring or fall.)
- Fertilize, water, and mow. You gotta mow at least a couple times a week to maintain the lower height since you don't want to cut more than 1/3 the length of grass at a time.
Additionally you'll also want to level and over seed, all steps might not be needed if you do the top list at the same time:
- Cut as short as you feel comfortable.
- Dethatch
- Aerate heavily then rake up and dispose of the plugs.
- Top dress with masons sand (fine sand but not powdery sand), then use a drag mat and push broom to work the sand into the canopy. Keep to a thin layer of sand so it doesn't smother the grass.
- Over seed