Answer: This weed is most rampant in rich moist soil. It moves fastest on bare soil so I suppose you could progressively kill off the lawn and allow it to colonize and I suppose you could transplant clumps of it to help it spread faster, however I would caution you that it is difficult keep in bounds and will encroach very well all on its own.
In all honesty, you might be happier in the long run if you planted one of the more traditional ground covers such as vinca minor (creeping myrtle) or English ivy (also available in variegated forms if you "hate" ivy) or even pachysandra. There are other shade loving plants you might use as a mass planting too such as woodland phlox, ferns, and the many forms of hostas (some of which are stoloniferous and spread nicely like a groundcover). These are all no-mow lawn substitutes and will not offend your neighbors.
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