Open AccessCoarse properties of graphs
Author(s) -
Tom Kaiser
Publication year - 2021
Language(s) - English
Resource type - Dissertations/theses
DOI - 10.35662/unine-thesis-2852
Subject(s) - mathematics , cayley graph , quotient , combinatorics , vertex transitive graph , multiplicative function , symmetric graph , voltage graph , graph , transitive relation , discrete mathematics , line graph , mathematical analysis
The objective of this work is to study large scale properties of graphs, graph sequences and groups. Firstly we consider the combinatorial cost and prove that having cost equal to 1 and hyperfiniteness are coarse invariants of graph sequences. We show that cost is multiplicative with respect to taking finite index subgroups. For an amenable group we investigate the properties of their Farber sequences, sofic approximations and vice versa. Secondly we consider the first uniformly finite homology group of graphs with coefficients in Z. We show that its non-vanishing depends on the ends, large circuits and (higher-dimensional) non-expansion of the graph. When the graph is transitive this is a full description. Finally we take a look at the Baumslag-Solitar group BS (2; 3). This group is non-Hopfian, meaning it has a quotient isomorphic to itself. We give a visual interpretation of this on the Cayley graph level.