Today’s post arises from a paper by my Penn State colleague Dima Burago and his collaborators, *Conjugation-invariant norms on groups of geometric origin.* (October 7, 2007). http://arxiv.org/abs/0710.1412.

The basic definition is a very simple one: say that a group \( G \) is *bounded * if it has finite diameter with respect to any bi-invariant metric. (Of course it is enough to consider the distance from a group element \( g \) to the identity element, and this function \( |g| \) then becomes a emph{conjugation invariant norm} on the group, hence the title of the paper. A group is then bounded if every conjugation-invariant norm is bounded from above. One can also ask whether every conjugation-invariant norm is bounded from below (away from the identity element). When a norm is bounded from above and below and can be said to be *trivial*: it is equivalent to the norm which equals 1 on all non-identity elements. If every conjugation-invariant norm is trivial the group is called *meager*.