This thesis concerns the actions of countable groups and associated Schreier graphs. In Chapters 1 and 2 we give the motivation and overview for the research presented in this thesis and we establish the basics regarding group actions, especially about Schreier graphs and amenability. Furthermore, we recall the idea of equationally compact actions of groups defined by Banaschewski. Finally, we show two results about equationally compact subgroups of infinite groups which answer two questions of Rajani and Prest.
We start off Chapter 3 with recalling the construction of a space of rooted Schreier graphs which are associated with the actions of a group. A crucial notion related to the space of rooted Schreier graphs is that of a Benjamini-Schramm convergence of sequences of sparse graphs, which has connections with measure-preserving actions of groups. We are, however, particularly interested in actions which only preserve the measure class, i.e. the non-singular actions of groups. Let us notice that for such actions the classical theorem of Radon-Nikodym can be applied, which equips the graph structure on the space with an additional function on the edges which forms a cocycle.
Thus, drawing inspiration from the space of rooted Schreier graphs, we construct a space of rooted Schreier cocycles of a group. Similarly as in the measure-preserving case, we obtain a correspondence between the space of cocycles and non-singular actions of groups.
In the final chapter of the thesis the central notion is that of hyperfiniteness, which has strong ties to amenability. The definition of hyperfiniteness varies between the settings of sequences of graphs, graphings of the actions of groups and for equivalence relations. Broadly speaking, an object is hyperfinite if it is in some sense close to being finite. Thus, a sequence of graphs is hyperfinite if we can remove sets of arbitrarily small size relative to the size of graphs in such a way that the resulting objects have components of bounded size. On the other hand, a measure preserving group action yields an associated structure of a graphing on the space that it acts upon. If we
can remove an arbitrarily small set from the probability space in such a way that the resulting graphing has bounded components then we call the action hyperfinite. In fact, these two notions of hyperfiniteness are strongly connected: by the theorem of Schramm, a measure preserving action is hyperfinite if and only if a sequence of graphs convergent to it is hyperfinite.
We consider a weighted version of hyperfiniteness, one which is suitable for this setting and we obtain a similar result to that of Schramm’s in Chapter 4, namely that a limit action of a hyperfinite sequence of cocycles is hyperfinite. Finally, we find continuous actions which are isomorphic to a given Borel action and have the same Radon-Nikodym cocycle and we obtain examples of free continuous actions of exact groups with continuous Radon-Nikodym derivatives.