Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H ( Γ ) and H + ( Γ ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space Cld F ∗ ( Γ × Γ ) of all closed sets of Γ × Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications H ¯ ( Γ ) and H ¯ + ( Γ ) . In this paper, we completely determine the topological type of the pair ( H ¯ + ( Γ ) , H + ( Γ ) ) and give a necessary and sufficient condition for this pair to be a ( Q , s ) -manifold. The pair ( H ¯ ( Γ ) , H ( Γ ) ) is also considered for simple examples, and in particular, we find that H ¯ ( T ) has homotopy type of R P 3 . In this investigation we point out a certain inaccuracy in Sakai–Ueharaʹs preceding results on ( H ¯ ( Γ ) , H ( Γ ) ) for finite graphs Γ.