No, it doesn't grow faster, thicker or otherwise more when you shave or cut it. Hairs are dead tissue -- the follicle in the skin cannot "know" what has happened to the rest of the hair.
What happens when you cut hair very short is that you are cutting hairs in all stages of their cycle. This includes those which are growing fast as well as those which have stopped at full length.
When you cut it longer, more of the hairs have already reached their final length, or are near it and so are growing slower. There are other hairs which are growing fast, but these are not yet long enough to be those you've cut.
Shaved hairs will of course continue to grow from that day, only stopping when they would have been full length anyway.