A variation on the unique product property of groups is described which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the group rings have no zero divisors. Some of its closure properties are described. It is shown that most surface groups satisfy this condition, and various other examples are given. It is explained how these ideas can give a more geometric interpretation of Promislow's example of a non-unique-product group.