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Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=d p (x,y) and aquasi-axis if lim infd(x,y)/d p (x,y)>0 asx, y range over the vertex set of p andd p (x,y)?8. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if a is a translation whose directionD(a) is a thin end, then there exists an axis inD(a) andD(a-1) invariant under ar for somer not exceeding the maximum number of disjoint rays inD(a).The thinness ofD(a) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.

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