We contribute some information towards finding a general algorithm for constructing, for a given profinite group, $G$, a compact connected space, $X$, such that the full homeomorphism group, $H(X)$, with the compact-open topology is isomorphic to $G$ as a topological group. It is proposed that one should find a compact topological oriented graph $\Gamma$ such that $G\cong Aut(\Gamma)$. The replacement of the edges of $\Gamma$ by rigid continua should work as is exemplified in various instances where discrete graphs were used. It is shown here that the strategy can be implemented for profinite monothetic groups $G$.
One knows that the compact homeomorphism group H(X) of a Tychonoff space has to be profinite ( [16], [17]). In the converse direction Gartside and Glyn [8] have established that every metric profinite group is the homeomorphism group of a continuum (i.e. a compact connected metric space).
For the goal of representing a given group as the homeomorphism group of a space, authors have pursued the following strategy:
Step (1): find some connected graph Γ, usually oriented, and find an isomorphic representation π: G → Aut(Γ); the standard attempt is to use some form of Cayley graphs (see [9], [8], [4])
Step (2): find a rigid continuum C, that is, a continuum, that is, compact connected metric space, whose only continuous selfmaps are the identity and the constant function (see [7], [11]) and replace each of the directed edges of Γ by C or a variant obtaining a connected space X; finally obtain an isomorphism γ: Aut(Γ) → H(X) (see [9], [11], [4]). Obtain an isomorphism γ • π: G → H(X).
All known variations of the strategy are highly technical, and different variations lead to rather different phase spaces X. It would be nice to find a construction which is in some way canonical, perhaps even functorial. However, one of the major obstructions for representations of a profinite group in a combination with graph theoretical methods is that homeomorphism groups, like all automorphism groups in a category are, in no visible way, functorial.
We propose, that in Step (1) one should in fact go more than halfway and construct a compact connected directed graph Γ and then apply Step (2) to achieve the final goal.
In the following we show that such constructions are possible in principle and yield for every profinite monothetic group G a continuum X such that H(X) ∼ = G. while not all compact monothetic groups are metric, the profinite ones among them are.
Thus, in the vein of a general existence result, our construction yields nothing new beyond what Gartside and Glyn have shown in [8]. However, the continua we construct are completely different from those produced in [8] and the proposed construction may turn out to be useful in the future.
In order to construct topological spaces with prescribed homeomorphism groups we first construct directed topological graphs with prescribed automorphism groups.
A directed (topological) graph is a triple Γ = (V, E, η) consisting of topological spaces V and E and a continuous function
The set V is called the space of vertices and E is called the space of (oriented) edges. If e ∈ E we write η(e) = (e 1 , e 2 ) ∈ V × V , then e 1 is the origin and e 2 is the target of e. Condition ( †) says
and this means that there is no vertex that is not an endpoint of an edge. Note that we allow a whole space η -1 (v 1 , v 2 ) of (directed) edges from v 1 to v 2 . We shall, however, not use this fact in the sequel.
If the spaces V and E of a directed graph Γ = (V, E, η) are discrete, we recover the more classical concept of a directed graph.
Example 2.2. (i) Let n be natural number n > 2 and let Z(n) = Z/n•Z be the cyclic group of n elements. Define
It is the Cayley-graph of the pair (Z(n), {1 + nZ}) consisting of the cyclic group of order n and the singleton generating set containing the element 1 + nZ.
(ii) More generally, let G be a topological group and let g ∈ G. We set
(iii) Taking Z with the discrete topology we obtain the Cayley graph C(Z) of (Z, {1}), the chain Z with its natural order-orientation.
⊓ ⊔ Definition 2.3.
A doubly pointed connected topological space Indeed we let X = E × L and define an equivalence relation ρ on X with the following equivalence classes:
If L is a Tychonoff link and F : L → I a morphism of links then our construction obviously induces a morphism F * : Γ L → Γ I of topological realisations. ⊓ ⊔ Notice that in the case of a Cayley graph of a group G with an element g ∈ G, the quotient space ((E × L))/ρ can be expressed in the form
and that there is a morphism
The verification of the details of the following examples is straightforward. Let (X, x 0 ) be a compact connected pointed space and (R, b 1 , b 2 ) a doubly pointed de Groot-continuum. Assume that X and R are disjoint with the exception of b 2 and x 0 which are assumed to be equal. Then a continuous function f : R → X ∪ R is exactly one of the following kind
Proof.
The function π: R ∪ X → R defined by π(X) = {b 2 } is continuous; hence the continuous self-map π • f : R → R is either the identity or is constant with image
) be a doubly pointed de Groot-continuum and F : R → I a morphisn of links. Then
From ( * ) above recall that for a cyclic group Z = Z/mZ, n = 0, 1, . . . we have |C(Z)| R = (Z × R)/ρ and note via Lemma 2.6, that the action (n, (k + mZ, x))
We now generalize Examples 2.7(i) to monothetic compact groups by utilizing Example 2.7(ii).
Let G be a compact group with a nonidentity element g. Let C(G) be the topological Cayley graph of (G, g). Proof.
(i) The assertions on
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