Liriope muscati is a large, clumping grasslike perennial that's traditionally been popular as a groundcover, due to its quick growth habit and spikes of light purple blooms. It's also known as big blue lilyturf, border grass, or lilyturf.
Though not actually spreading by underground stems, an individual plant after several years can reach 24 inches in width by suckering at the base and will quickly cover an open area.
It's that spreading growth habit that has moved the UF/IFAS Assessment of Non-Native Plants in Florida's Natural Areas to list Liriope muscari (and the related Liriope spicata) as a high invasion risk. UF/IFAS no longer recommends using liriope in the landscape.
If you're looking for an alternative border grass, mondo grass might be an option.