Groups acting on graphs with polynomial growth (Q1175422)

Let \(X\) denote an infinite, locally finite connected graph, and let \(G\) be a finitely generated group acting on \(X\) with finitely many orbits. If \(G\) is nilpotent, then every element of \(G\) of infinite order fixes no non-empty finite subset of \(V(X)\); if \(G\) has polynomial growth degree \(d\geq 1\), then all vertex-stabilizers are finite and \(X\) has the same polynomial growth degree \(d\). If \(X\) has polynomial growth but \(G\) acts transitively with nonpolynomial growth, then Aut(\(X\)) is uncountable. \textit{M. Gromov} [Publ. Math., Inst. Hautes Etud. Sci. 53, 53-78 (1981; Zbl 0474.20018)] proved that finitely generated groups with polynomial growth contain a normal nilpotent subgroup of finite index. The author shows that if Aut(\(X\)) is such a group, then \(X\) is contractible to a Cayley graph of a nilpotent group with polynomial growth. Suppose \(X\) has polynomial growth degree \(d\). It is conjectured that Aut(\(X\)) is uncountable if and only if it contains a finitely generated subgroup \(G\) with exponential growth. The conjecture is proved when \(X\) is a strip \((d=1)\); here \(G\) must be metabelian with two generators.