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A survey on graphs with polynomial growth
AbstractIn this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: •Automorphism groups of graphs with polynomial growth.•Groups and graphs with linear growth.•S-transitivity.•Covering graphs.•Automorphism groups as topological groups
Covering theory for complexes of groups
We develop an explicit covering theory for complexes of groups, parallel to
that developed for graphs of groups by Bass. Given a covering of developable
complexes of groups, we construct the induced monomorphism of fundamental
groups and isometry of universal covers. We characterize faithful complexes of
groups and prove a conjugacy theorem for groups acting freely on polyhedral
complexes. We also define an equivalence relation on coverings of complexes of
groups, which allows us to construct a bijection between such equivalence
classes, and subgroups or overgroups of a fixed lattice in the
automorphism group of a locally finite polyhedral complex .Comment: 47 pages, 1 figure. Comprises Sections 1-4 of previous submission.
New introduction. To appear in J. Pure Appl. Algebr
2-Arc-transitive metacyclic covers of complete graphs
Regular covers of complete graphs whose fibre-preserving automorphism groups act 2-arc-transitively are investigated. Such covers have been classified when the covering transformation groups K are cyclic groups Z(d) for an integer d >= 2, metacyclic abelian groups Z(p)(2), or nonmetacyclic abelian groups Z(p)(3) for a prime p (see S.F. Du et al. (1998) [5] for the first two metacyclic group cases and see S.F. Du et al. (2005) [3] for the third nonmetacyclic group case). In this paper, a complete classification is achieved of all such covers when K is any metacyclic group. (C) 2014 Elsevier Inc. All rights reserved.116Ysciescopu
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