# Some orders on groups

A left order on a group is a total order of the underlying set such that left multiplication by any element of the group is monotonous. Dually we have the definition of right order. A bi-order is an order that is both a left and a right order.

Example 1: The usual order on the integers is a bi-order for the usual group structure.

Example 2: The same is true for the real numbers.

Problem 1: Give a bi-order in the free group generated by an arbitrary set.

Now we will give a very strong consequence of having a right order. First we start with a nice excercise.

Problem 2: Find an infinite countable discrete subset of the real numbers such that the induced order in this set does not have maximum nor minumum and such that for any pair of distinct elements there is a third element in between.

We are ready to state the following very nice result:

Problem 3: Let $B$ be a triangulated locally finite space. Suppose given a right order in $\pi_1(B)$. Let $p : E\to B$ be the universal cover of $B$. Then there exists $h : E \to \mathbb{R}$ such that $(p,h) : E \to B \times \mathbb{R}$ is an embedding.

In particular the universal cover of $B$ embeds in $B\times \mathbb{R}$. Compare this with the classical drawing of the universal cover of $S^1$!

## 2 thoughts on “Some orders on groups”

1. Hint for 1: Start by finiding an order for the free (non-commutative) power series on the given set, that is compatible with addition. Find an order that is also compatible with multiplication when restricted to the units of the ring.

Hint for 2: Consider the complement of some set.

Hint for 3: This might be hard.

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2. […] that left multiplication by any element of the groups is monotonous (for very simple examples see Some orders on groups). Define an element of an ordered group to be positive if it is greater than the unit of the group. […]

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