They have a thin elastic band but since I bought a size up it doesn't pucker or move at all. I wear them to bike lessons and they don't ride up. They absorb moisture. My inhaler fits in a tiny pocket and doesn't move. In size XXL I would describe them as medium rise with a little more room below the waist. UPDATE 2/25/19: The style above is #6 - I just ordered style #10 and I like them much more, they have a higher waist, wider waistband and slightly shorter legs. There is no floating rubber.
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