CEU Electronic Theses and Dissertations, 2018
| Author | Tóth, László Márton |
|---|---|
| Title | Limiting Techniques in Measured Group Theory |
| Summary | Limits of finite structures is a topic that has been rapidly developing over the last two decades. The common theme of building limiting theories has been present in probability theory, extremal graph theory and group theory, just to name the areas closest to the scope of this thesis.The shared philosophy of all these areas is that the limiting language creates a connection between the combinatorial nature of the finite structures and the analytic properties of the limiting objects. In many cases the tools available for the finite and infinite worlds reinforce each other and lead to surprising results, new notions and interesting questions. In this thesis we exhibit examples of this approach in the field of measured group theory, where the main objects of interest are probability measure preserving (p.m.p.) actions of groups. These arise naturally as limiting objects in the sparse graph limit theory. First we investigate invariant random subgroups (IRS's) in groups acting on rooted trees. We prove that a nontrivial ergodic IRS of the group of alternating finitary automorphisms that acts without fixed points on the boundary of the tree contains a level stabilizer. In particular it is the random conjugate of a finite index subgroup. We generalize this to certain branch groups, and also to the case when fixed points are allowed. We then procede to show a connection between the notions of cost (by Gaboriau) and combinatorial cost (by Elek), and as an application prove new results for the rank gradient of finitely generated groups. In the last chapter we investigate the distortion function of p.m.p. actions. This invariant was recently introduced by Abért, Gelander and Nikolov to measure the necessary complexity of generation when getting close to the cost. We compute the distortion function for free actions of $Z^d$, and show a logarithmic upper bound for a wide generality of lamplighter groups. |
| Supervisor | Abért, Miklós |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/toth_laszlo-marton.pdf |
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