Representing a profinite group as the homeomorphism group of a continuum

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 $Γ$ such that $G\cong Aut(Γ)$. The replacement of the edges of $Γ$ 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 $G$.

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