I have loads of the big violets carpeting whole ares of the wild cedar forest behind my house. They don't have any scent, unfortunately but do a brilliant job as a ground cover. They do try to venture into the grassed areas nearby but (so far) not to any disturbing extend. Lovely to look at in the spring!
A lawn that isn't in deep shade and gets mowed regularly can outgrow violets, even the big V. sororia/cucullata/priceana/papillionacea or whatever they're called. They are lovely. I wish someone could breed scent into that tribe, since they're fiercely hardy and grow like weeds.
Thanks for your comment dawnsharron - and you certainly called it a good few names!!
Next to the grass they are no problem indeed. I found some planted in the front bed by the house and am still removing the last stragglers years later! In the early spring they look a mess - just this bumpy, funny surface where the lush green growth had been. then seemingly overnight the carpet is back! I have a race on in my forest - periwinkle versus violets. Not too keen for a mix though!