Combine 2 cups water, sugar, and the gingerroot in a saucepan set over high heat. Bring to a boil, stirring constantly to dissolve the sugar, and then reduce heat to a simmer. Simmer for 3 minutes.
Strain through a chinois set over a bowl, and cool completely.
Combine the ginger syrup, remaining water, and the lime juice together in a pitcher. Stir well.
Serve over ice, and garnish the glasses with lime slices.