We describe a variation on the unique product property of groups, which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the groups rings have no zero divisors. We describe some of its closure properties under extentions and amalgamated free products. We show that most surface groups satisfy this condition, and give various other examples. We explain how these ideas can give a more geometric interpretation of Promislow's example of a non-u.p. group.
J. London Math. Soc. 62 (2000) 813-826.