Finite contractions of graphs with polynomial growth
type: Item
instance of: scholarly article
title: Finite contractions of graphs with polynomial growth
zbMATH Open document ID: 0964.05028
author: András Lukács
author: Norbert Seifter
published in: European Journal of Combinatorics
publication date: 2001-07-08T00:00:00Z
review text: 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 \(X\).
reviewed by: Mark E. Watkins
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 1567137
zbMATH Keywords: Hirsch number
zbMATH Keywords: Cayley graph
zbMATH Keywords: nilpotent group
zbMATH Keywords: almost nilpotent group
zbMATH Keywords: imprimitivity system
MaRDI profile type: Publication
full work available at URL: eujc.2000.0405
OpenAlex ID: W2122744904
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DOI: 10.1006/EUJC.2000.0405
description: scientific article; zbMATH DE number 1567137