In Sub-riemannian geometry and Lie groups , Part I, section 4.2, I introduced the strange notion of “uniform group”. Instead of saying that an uniform group is just a topological group (which has an unique uniformity associated to it), I proposed the following construction.
1. Double of a group. To any group we associate its double group with the group operation
I introduce also the following three functions:
, (where is the neutral element of the group ),
Remark that all these functions are group morphisms. Notice especially the morphism , which is nothing but the group operation of , seen as a group morphism from to .
2. Uniform group. For my purposes I introduced the notion of an uniform group: an uniform group is a group , together with two uniformities, one on , the other on , such that the three morphisms from the point 1. are uniformly continuous.
So, instead of one uniformity, now I use two. Why? Well, we may eliminate one of the uniformities, the one on . Indeed, suppose that is an uniform group. Then take on the smallest uniformity which makes the three morphisms uniformly continuous. Now we have two uniformities, one on the double of , the other on the double of the double of . We may repeat the procedure indefinitely, pushing to infinity the pair of uniformities.
(TO BE CONTINUED)