Efficient construction of the reals.

 October 20, 2006 mathematics

Today in Geometry/Topology seminar, quasihomomorphisms \mathbb{Z}\to \mathbb{Z} were discussed, i.e., the set of maps f : \mathbb{Z}\to \mathbb{Z} such that | f(a+b) - f(a) - f(b) | 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 \mathbb{R} , 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 d on the group), I can define a quasihomomorphism f : G \to G by demanding d(f(ab),f(a)f(b)) be uniformly bounded, and where two quasihomomorphisms f, g are equivalent if d(f(a),g(a)) is uniformly bounded. Let’s call the resulting object \hat{G} for now.

What can be said about \hat{G} ? For instance, what is \hat{F_2} ?