Topological groups and infinite graphs

AbstractWe show how results concerning infinite, locally finite, vertex-symmetric graphs can be related with the structure theory of topological groups, when the latter is applied to automorphism groups of the graphs. In particular, we discuss polynomial growth, bounded automorphisms and infinite expanders. In an appendix, we present three problems on infinite graphs, not necessarily linked with topological considerations

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

Percolation and isoperimetry on roughly transitive graphs

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When G is transitive, the fact that p_c < 1 was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov's theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.Comment: 32 pages, 2 figure