  An asymmetric coloring of a graph is a coloring of its vertices that is not
preserved by any non-identity automorphism of the graph. The motion of a graph
is the minimal degree of its automorphism group, i.e., the minimum number of
elements displaced by any non-identity automorphism. In this paper we confirm
Tom Tucker's "Infinite Motion Conjecture" that connected locally finite graphs
with infinite motion admit an asymmetric 2-coloring. We infer this from the
more general result that the inverse limit of a sequence of finite permutation
groups with disjoint domains, viewed as a permutation group on the union of
those domains, admits an asymmetric 2-coloring. The proof is based on the study
of the interaction between epimorphisms of finite permutation groups and the
structure of the setwise stabilizers of subsets of their domains.
