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If $X$ is a locally finite graph, (i.e. each vertex has finite index), is it true that the automorphism group Aut($X$) of the graph X is locally compact?

Here, Aut($X$) has compact open topology; and topology of $X$ is the weak topology when we consider $X$ as a CW-complex (see. Hatcher, Algebraic Topology - Graphs and Free Groups).

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    The question refers to automorphisms which are linear in the edges? Since you say "we consider X as a CW-complex", by Aut(X) I would understand the group of self-CW-homeomorphisms of X — but even for the connected graph with two vertices, also known as [0,1], the group of self-CW-homeomorphisms is not locally compact.2011-07-04
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    @Mariano: presumably the OP means graph automorphisms since, as you say, the result is not otherwise true.2011-07-04

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