TITLE: Unbounded metrics on groups
QUESTION [6 upvotes]: If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?

REPLY [8 votes]: The answer is no. See Thm 1.2 of http://homepages.math.uic.edu/~rosendal/PapersWebsite/Property(OB)10.pdf. There is a property discussed in the intro of this paper which is equivalent to all left invariant metrics are bounded. It is known that certain large permutation groups have this property.