The group of germs at infinity of line homeomorphisms

Recently, I learned that the group G_∞ of germs at +∞ of orientation preserving homeomorphisms of the real line has two remarkable properites: it is simple and left-orderable. I thought it is maybe worth to share some background on left-orderability (and why this might be interesting). Let’s start with a definition:

A group G is left-orderable, if there is a total order ≺ on G which is preserved by left multiplication, i.e., for all elements a,b,g

\[a \prec b \Longrightarrow ga \prec gb\]

Not every group is left-orderable. For example, it is rather clear that a group with a left-order is torsion-free. Indeed, if a is an element, say 1 ≺ a, then multiplying with a from the left one has 1 ≺ a≺ a²≺ a³≺ … and so on, i.e., a has infinite order. Nevertheless, in general it can be difficult to decide whether a group is left-orderable. (Remarkably, there is an algorithm which takes as input a countable group with solvable word problem and stops in finite time exactly if the group is not left-orderable.) Orderability seems more algebraic than it essentially is. For countable groups there is a very neat dynamical characterization of orderability:

Theorem: A countable group is left-orderable if and only if it acts faithfully by orientation preserving homeomorphisms on the real line.

This is nicely written up in E. Ghys’ article “Groups acting on the circle” (L’Enseignement Math. 47, 2001). In other words, every countable left-orderable group is a subgroup of Homeo₊(ℝ).

Let me indicate the argument. Why is a countable subgroup G of Homeo₊(ℝ) left-orderable? The group G is countable, so there is a countable set of points x₁,x₂,x₃,… in ℝ which can distinguish any pair of distinct elements in G.
Now a suitable lexicographical order can be defined:

\[ f \prec g \Leftrightarrow \exists i f(x_i) < g(x_i) \text{ and } f(x_j) = g(x_j) \text{ for all j < i} \]

The converse direction is slightly more delicate. Essentially, one identifies the ordered group in an order-preserving way with suitable subset of ℝ and extends the left-multiplications to continuous functions on ℝ.

Let’s turn this around: Are there left-orderable groups which don’t act faithfully on the line? Of course the answer is yes: the cardinality of Homeo₊(ℝ) is the continuum and it is easy to find left-orderable groups of arbitrary cardinality. So, the remaining interesting question might be: Are there left-orderable groups of continuum cardinality which don’t act on the line? The answer is again yes and the group G_∞ of germs at +∞ of homeomorphisms of the line is one such example. Details can be found in an article of Kathryn Mann.

Leave a comment, if you know other examples of “groups of germs at infinity” with remarkable properties.