Answer: Most groundcovers spread by underground roots, so your choice is limited to plants that sprawl from a central stem rather than those that multiply by producing rhizomes or stolons. How about heather, trailing euonymus, houttuynia, or sunrose for your sunny slope? Euonymus will root through the landscape fabric (or attach itself to it). Once your groundcover is established, it should be watered deeply during the hottest months of the year, but it will take far less water than a lawn. Enjoy!
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