TY - BOOK

T1 - Cantor-Bendixson type ranks & Co-induction of invariant random subgroups

AU - Quorning, Vibeke

PY - 2018

Y1 - 2018

N2 - The present thesis consists of two unrelated research projects and is therefore divided into two parts. The rst part is based on the paper [27]. The second part is based on the paper [22], which is joint with Alexander S. Kechris. Part I. For a Polish space X it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on F@0 (X) if and only if X is -compact. We construct a family of co-analytic ranks on F@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 -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 F2 which are all invariant and weakly mixing with respect to the action of Aut(F2). Finally, by studying continuity properties of our operation, we obtain results concerning the continuity of the coinduction operation for weak equivalence classes of measure preserving group actions.

AB - The present thesis consists of two unrelated research projects and is therefore divided into two parts. The rst part is based on the paper [27]. The second part is based on the paper [22], which is joint with Alexander S. Kechris. Part I. For a Polish space X it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on F@0 (X) if and only if X is -compact. We construct a family of co-analytic ranks on F@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 -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 F2 which are all invariant and weakly mixing with respect to the action of Aut(F2). Finally, by studying continuity properties of our operation, we obtain results concerning the continuity of the coinduction operation for weak equivalence classes of measure preserving group actions.

M3 - Ph.D. thesis

BT - Cantor-Bendixson type ranks & Co-induction of invariant random subgroups

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -