Ditto on liriope.
Can't kill it. The running super aggressive type can just be left on top of the ground and will still grow.
I've seen people treat it like turf... mowing it every time they mowed the lawn... didn't bother it.
Just be aware that most of these 'ground covers' are super aggressive and often escape cultivation and wreak havoc over native ecosystems.
Personally, I'd just bring in a nice woodchip mulch and then gradually sprig in nice plants like the native ferns in the link posted...
The woodchips look nice and aren't really bothered by the dog traffic... Although... the armadillo root around in them in my area.
