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≺ a ^{2}≺ a^{3}≺ …* 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 x

_{1},x

_{2},x

_{3},… 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.

Congratulations to your first post on this blog, Steffen! And also ‘thank you’ – any help is much appreciated.