Influence of the structure of a graph on its automorphisms (Q1813169)

Let \(G\) be a connected, locally finite, vertex transitive graph, \(x\) a vertex of \(G\), and \(g\) an automorphism of \(G\). In an earlier paper [Mat. Sb. (n.S) 134, No. 2, 274-284 (1987; Zbl 0698.05035)], the author studied the asymptotic behaviour of the function \(\xi_{G,x,g}(n)=\max\{\hbox{dist}_ G(y,g(y));\) \(y\in V(G)\) such that \(\hbox{dist}_ G(x,y)\leq n\}\). The present paper is devoted to investigating the effect of both the growth of \(G\) and the growth of \(\xi_{G,x,g}\) on the group \(\hbox{Aut}(G)\). As the main result, the following theorem is proved. If, for some \(a\), \({1\over 2}\leq a<1\), one has \(\hbox{ln}|\{y;\hbox{dist}_ G(x,y)\leq n\}|=O(n^ a)\) and, at the same time, \(\xi_{G,x,g}(n)=O(n^{1/a-1})\), then there is an imprimitivity system \(s\) of the group \(\hbox{Aut}(G)\) on \(V(G)\) with finite blocks such that the group \(\langle\hbox{Cl}_{\hbox{Aut}(G)}(g)\rangle^ s\) is a finitely generated nilpotent group. Some consequences of the result are discussed.