1 Groups
In this chapter, we prove Holder’s theorem for groups. We follow the proof in “Groups, Orders, and Dynamics” by Deroin, Navas, and Rivas.
We choose an element of the ordered group \(f\) and map it to \(1\) in the real numbers. Then for any other element \(g\) of the ordered group, we use \(f\) to construct a sequence of rational approximations of \(g\). We map \(g\) to the real number that is the limit of this sequence of approximations. We then prove that this map is injective and order preserving.
1.1 Definitions
We begin with basic definitions of ordered groups.
A left ordered group \(G\) is a group and a partial order such that for all \(x, y, z \in G\), if \(x \le y\), then \(z * x \le z * y\).
A left linear ordered group \(G\) is a left ordered group that is also a linear order.
An Archimedean group is a left ordered group such that for any \(g, h \in G\) where \(g \ne 1\), there exists an integer \(z\) such that \(h {\lt} g^z\).
1.2 Approximation
In this section we assume that \(G\) is a left linear ordered group that is Archimedean. Furthermore, we assume we have an element \(f\in G\) such that \(1 {\lt} f\).
For any \(g \in G\) and \(p \in \mathbb {N}\), there exists an integer \(q \in \mathbb {Z}\) such that
Since \(G\) is an Archimedean group, we can construct exponents \(l\) and \(u\) such that \(f^l {\lt} g^p {\lt} f^u\). Therefore, there must exist some integer \(q\) which satisfies what we want.
We define a function \(q \colon G \to \mathbb {N} \to \mathbb {R}\) using Theorem 4 such that for any \(g\in G\) and \(n \in \mathbb {N}\),
For any sequence \(a_n\) of real numbers, if there exists \(C \in \mathbb {R}\) such that for all \(m,n\in \mathbb {N}\) we have that
then sequence \(\frac{a_n}{n}\) converges.
Not included here as the ideas are separate from this project.
For any \(g \in G\) and \(a,b \in \mathbb {N}\), we have that
We know the following two things by the definition of \(q\)
And so it follows that
For any \(g \in G\), the sequence \(\frac{q_g(n)}{n}\) converges.
1.3 Map
We make the same assumptions as in the previous section. So we assume that \(G\) is a left linear ordered group that is Archimedean, \(f\in G\), and \(1 {\lt} f\).
We now define the map from the \(G\) to \(\mathbb {R}\) and prove its properties.
We define a map \(\phi \colon G \to \mathbb {R}\) by mapping \(g\) to the real number that \(\frac{q_g(n)}{n}\) converges to as we know from Theorem 8.
For all \(g_1,g_2 \in G\) and \(p\in \mathbb {N}\),
Let \(q_1 = q_{g_1}(p)\) and \(q_2 = q_{g_2}(p)\). Then we know that
And so we also have the following two facts
We look at the case where \(g_1g_2 \le g_2g_1\). Then \(g_1^p g_2^p \le (g_1g_2)^p \le g_2^p g_1^p\). And so combined with the previous facts, we have that
Therefore,
And so we are done. The case where \(g_2g_1 \le g_1g_2\) follows similarly.
The map \(\phi \) is a homomorphism.
Let \(g_1, g_2 \in G\). Then from Theorem 10 we have that
And so since \(\lim _{p\to \infty }\frac{q_{g_1}(p) + q_{g_2}(p)}{p} = \lim _{p\to \infty }\frac{q_{g_1}(p) + q_{g_2}(p) + 1}{p}\), we have that
Therefore, by the definition of \(\phi \), we have shown that \(\phi (g_1)+\phi (g_2) = \phi (g_1g_2)\).
For all \(g,h \in G\), if \(g \le h\) then \(\phi (g) \le \phi (h)\).
First, notice that since \(g\le h\), then for all \(p \in \mathbb {N}\), \(q_g(p) \le q_h(p)\). Then from the definition of \(\phi \), it follows that \(\phi (g) \le \phi (h)\).
We have that \(\phi (f) = 1\) where \(f\) is our fixed positive element.
We have that for all \(n\in \mathbb {n}\) that \(f^n \le f^n {\lt} f^{n+1}\) and so \(q_f(n) = n\). Therefore, \(\phi (f) = 1\).
The map \(\phi \) is injective.
Since from Theorem 11 we have that \(\phi \) is a homomorphism, it suffices to show that for any \(g\in G\), if \(\phi (g) = 0\), then \(g = 1\).
Assume for the sake of contradiction that there exists \(g \in G\) such that \(\phi (g) = 0\) but \(g\) is not equal to \(1\). Then since \(G\) is Archimedean, there exists an integer \(z\) such that \(g^z {\gt} f\). Therefore, since by Theorem 13 we have that \(\phi (f) = 1\),
Contradiction.
For all \(g,h \in G\), we have that \(g \le h\) if and only if \(\phi (g) \le \phi (h)\).
(\(\Rightarrow \)) This is Theorem 12.
(\(\Leftarrow \)) We have that \(\phi (g) \le \phi (h)\). Assume for the sake of contradiction that \(h {\lt} g\). Then by Theorem 12, we know that \(\phi (h) \le \phi (g)\). Therefore, \(\phi (g) = \phi (h)\). And so by Theorem 14, we know that \(\phi \) is injective and so \(g = h\). Contradiction.
1.4 Holder’s Theorem
If \(G\) is a left linear ordered group that is Archimedean, then \(G\) is isomorphic to a subgroup of \(\mathbb {R}\).
First, we look at the case where there exists a positive element \(f\) in \(G\). Given such an element, we have an order preserving, injective homomorphism \(\phi \). And so \(G\) is isomorphic to the image of \(\phi \) which is a subgroup of \(\mathbb {R}\).
If there does not exist a positive element in \(G\), then \(G\) is trivial and is isomorphic to the trivial subgroup of \(\mathbb {R}\).