The automorphism group of the real line with standard topology

Solution 1:

As I said in my comments, there are many groups which embed (as subgroup) in the group $G=Homeo_+(R)$, the group of orientation-preserving homeomorphisms of $R$ (which is index $2$ subgroup in $Homeo(R)$). Whether these groups qualify as interesting is the matter of taste, since some people find only finite groups to be interesting (and among finite groups only the trivial group embeds in $G$).

To begin with, a countable group is left orderable if and only if it embeds in $G$. (See here for further details on orderability of groups.) It is known that all locally indicable groups, all braid groups, all (finitely generated) right angled artin groups, all residually torsion-free nilpotent groups, embed in $G$. For something more concrete: all finitely generated free groups, all surface group embed in $G$, the fundamental group of each compact 3-dimensional hyperbolic manifold contains a finite index subgroup which embed in $G$, same for the fundamental groups of all knot complements. See this very recent preprint for proofs and references.

Addendum: Topology of $G$, which is a topological group when equipped with topology of uniform convergence on compacts.

I added it here in order to justify the guess made by Oliver Begassat:

Lemma. The group $G$ is contractible.

Proof. Let $G_0<G$ denote the subgroup of homeomorphisms fixing the origin. Then the formula $$ F(x,t)=f(x) - tf(0) $$ defines a homotopy of every $f\in G$ to $f_1=F(x,1)\in G_0$, thereby establishing that the inclusion $G_0\to G$ is a homotopy equivalence.

For $f\in G_0$ define the homotopy $$ H(x,t)= (1-t)f(x) + tx. $$ It s elementary to see that for every fixed $t$, the map $H(\cdot, t)$ is continuous, strictly increasing and surjective, as a map ${\mathbb R}\to {\mathbb R}$. For $t=1$, $H(x,t)=x$. Therefore, we obtain that $G_0$ is contractible. qed

As for Ittay's concerns about torsion: Every nontrivial element of the group $G$ has infinite order.

Solution 2:

One very interesting subgroup is the set $$ \{ \phi \in Aut(\mathbb{R}) : \phi(x+1)=\phi(x)+1 \} $$

These are precisely those homeomorphisms which are the lifts of circle homeomorphisms. Circle homeomorphisms and their properties have been studied for some time in dynamical system theory.

Poincare started the study of circle maps when proved that the rotation number $$\alpha = \lim_{n\to\infty} \frac{\phi^n(x)-x}{n} $$ exists and is independent of the point $x$.