Finite contractions of graphs with polynomial growth
The Hirsch number \(h(G)\) of a polycyclic group \(G\) is the number of infinite cyclic factors in a descending series from \(G\) whose factors are cyclic or finite. A finite contraction \(H\) of a graph \(X\) is a graph whose vertices are pairwise disjoint, finite, connected subgraphs \(X_i\) of \(X\), and \([X_i,X_j]\) is an edge of \(H\) if in \(X\) some vertex of \(X_i\) and some vertex of \(X_j\) are adjacent. It is first shown that if \(X\) is any Cayley graph of a torsion-free, finitely generated nilpotent group \(G\), then the lattice of dimension \(h(G)\) is a finite contraction of a subgraph of \(X\). From this one obtains the main result: Let \(X\) be a vertex-transitive, connected, locally finite graph with polynomial growth. Then there exists a quotient group of Aut\((X)\) that contains a finitely generated nilpotent subgroup \(N\) with the same growth rate as \(X\) and the \(h(N)\)-dimensional lattice is a finite contraction of some subgraph of \(X\).
scientific article; zbMATH DE number 1567137