Efficient construction of the reals.

 2006-10-20 04:48:38 +0000 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 $\R$, 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 $\hat{F_2}$?