Today in Geometry/Topology seminar, quasihomomorphisms were discussed, i.e., the set of maps such that is uniformly bounded, modulo the relation of being a bounded distance apart. These come up when defining rotation and translation numbers, for instance.

Anyway, Uri Bader mentioned that these quasihomomorphisms form a field, isomorphic to , under pointwise addition and composition. I hadn’t realized that this is a general construction. Given a finitely generated group (with fixed generating set, so we have the word metric on the group), I can define a quasihomomorphism by demanding be uniformly bounded, and where two quasihomomorphisms are equivalent if is uniformly bounded. Let’s call the resulting object for now.

What can be said about ? For instance, what is ?