This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/31444 |
| Date | January 2014 |
| Creators | Cannizzo, Jan |
| Contributors | Kaimanovich, Vadim |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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