Fringe (Apron / Collar)
The narrow band of slightly longer grass bordering the green, between the green itself and the fairway or rough.
Which name you use is mostly regional: British golfers traditionally say "apron," Americans say "fringe," and "collar" shows up as a third synonym for the same thing. Under the actual Rules of Golf, none of the three names matter, this area is simply part of the general area, not the putting green.