Given a set $S$, the free group on $S$ consists of finite strings of elements in $S$. They can be visualized as paths on the integer grid $\mathbb{Z}^S$ starting at the origin, with the group operation given by path concatenation.
There is no set for which $\mathbb{R}$ is the corresponding free group- however it is tempting to view $\mathbb{R}$ as a free group on a single "infinitesimal generator". More generally given a set $S$ we could consider the group where
- elements are pairs $(T, f)$ with $T \in \mathbb{R}$ and $f: [0,T] \to \mathbb{R}^S$ a continuous function and $f(0)=0$. (The value of $T$ mirrors the length of a string in the free group. )
- the product of $(T_1,f_1)$ with $(T_2,f_2)$ is a map $[0,T_1+T_2]\to\mathbb{R}^S$ given by $f_1(t)$ for $t \in [0,T_1)$ and $f_1(T_1)+f_2(t-T_1)$ for $t\in[T_1, T_1 + T_2]$
- the identity element is $(0,0\mapsto 0)$,
In a commutative version of this definition, as in the free commutative group, two paths would be equal if they have the same start and end point.
Does this construction have a name? I see there is infinitesimal generators in the context of Lie groups, but that does not seem to be related...
Edit: the paths should also be equivalent up to reparameterization.