# Defense Vibeke Quorning

Title: Cantor-Bendixson Type Ranks & Co-Induction and Invariant Random Subgroups

Abstract:

The thesis consists of two unrelated research projects and is therefore divided into two parts. The first part is based on the paper Cantor-Bendixson type ranks, Vibeke Quorning, 2018. The second part is based on the paper Co-induction and invariant random subgroups, Alexander S. Kechris and Vibeke Quorning, 2018.

Part I. For a Polish space $X$ it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on $F_{\aleph_0}(X)$ if and only if $X$ is $\sigma$-compact. We construct a family of co-analytic ranks on $F_{\aleph_0}(X)$ for any Polish space $X$. We study the behaviour of this family and compare the ranks to the Cantor-Bendixson rank. The main results are characterizations of the compact and $\sigma$-compact Polish spaces in terms of this behaviour.

Part II. We develop a co-induction operation for invariant random subgroups. We use this operation to construct new examples of continuum size families of non-atomic, weakly mixing invariant random subgroups of certain kinds of wreath products, HNN extensions and free products with normal amalgamation. Moreover, by use of small cancellation theory together with our operation, we construct a new continuum size family of non-atomic invariant random subgroups of $\mathbb{F}_2$ which are all invariant and weakly mixing with respect to the action of $\text{Aut}(\mathbb{F}_2)$. Finally, by studying continuity properties of our operation, we obtain results concerning the continuity of the co-induction operation for weak equivalence classes of measure preserving group actions.

Supervisor:  Associate Professor, Asger Törnquist, MATH, University of Copenhagen

Assessment Committee:

Associate Professor,  (Chairman), Magdalena Musat, MATH, University of Copenhagen

Associate Professor, David Kyed, SDU IMADA, Denmark

Associate Professor, Julien Melleray, Universite Lyon, France