For ginger syrup, in a blender combine the water and ginger. Cover and blend until ginger is pureed. Transfer mixture to a small saucepan. Stir in sugar. Bring to boiling; reduce heat. Simmer, uncovered, about 10 minutes or until mixture is reduced to 1 cup; cool completely. Chill for at least 30 minutes (up to 1 week).
For each float, pour 1/4 cup of the ginger syrup into a tall glass; add 1 tablespoon of the lime juice and 1 torn mint sprig. To muddle, gently crush mint with the back of a spoon. Add 1/4 cup of the sorbet; slowly fill with club soda. If desired, garnish with lime peel.