Representing a profinite group as the homeomorphism group of a continuum

2011/08/19 by Hofmann, Karl H., Morris, Sidney A.
#2010: 22C05 #22F50 #54H15 #57S10 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1108.3876

Abstract

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≅ 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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