What type plant is this?
A: Monkey Grass

This has several names: lily turf or monkey grass. Its botanical name is liriope. It grows well in many conditions including wet soils. Some varieties are more aggressive and can escape from where the gardener wants it. TSU Extension Agent Joellen Dimond says it only needs to be cut once a year. Make sure to dispose of the clippings. This will keep it looking nice and reduce foliar diseases. 

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Related Resources:
Liriope Muscari
Liriope Culture