Full groups, classification, and equivalence relations

In Chapter I, we study algebraic properties of full groups of automorphisms of sigma-complete Boolean algebras. We consider problems of writing automorphisms as compositions of periodic automorphisms and commutators (generalizing work of Fathi [34] and Ryzhikov [69]), as well as problems concerning the connection between normal subgroups of a full group and ideals on the underlying algebra, in the process giving a new proof (joint with David Fremlin) of Shortt's [73] characterization of the normal subgroups of the group of Borel automorphisms of an uncountable Polish space, as well as a characterization of the normal subgroups of full groups of countable Borel equivalence relations which are closed in the uniform topology of Bezuglyi-Dooley-Kwiatkowski [9]. We also characterize the existence of an E-invariant Borel probability measure in terms of a purely algebraic property of [E].;The results of Chapter II include classifications of Borel automorphisms and Borel forests of lines up to the descriptive analog of Kakutani equivalence, along with applications to the study of Borel marriage problems, generalizing and strengthening results of Shelah-Weiss [72], Dougherty-Jackson-Kechris [24], and Klopotowski-Nadkarni-Sarbadhikari-Srivastava [58]. We also study the sorts of full groups on quotients of the form X/E for which the results of Chapter I do not apply. Actions of such groups satisfy a measureless ergodicity property which we exploit to obtain various classification and rigidity results. In particular, we obtain descriptive analogs of some results of Connes-Krieger [19] and Feldman-Sutherland-Zimmer [37], answering a question of Bezuglyi.;In Chapter III, we study some descriptive properties of quasi-invariant measures. We prove a general selection theorem, and use this to show a descriptive set-theoretic strengthening of an analog of the Hurewicz ergodic theorem which holds for all countable Borel equivalence relations. This then leads to new proofs of Ditzen's quasi-invariant ergodic decomposition theorem [23] and Nadkarni's [62] characterization of the existence of an E-invariant probability measure, and also gives rise to a quasi-invariant version of Nadkarni's theorem, as well as a version for countable-to-one Borel functions. We close Chapter III with results on graphings of countable Borel equivalence relations, strengthening theorems of Adams [1] and Paulin [65].