ON QUOTIENTS OF NONSINGULAR ACTIONS WHOSE SELF-JOININGS ARE GRAPHS

Abstract

We study the notions of minimal self-joinings (MSJ) and graph self-joinings (GSJ) (analogous to simplicity in the finite measure preserving case) for nonsingular actions of locally compact groups. We show that for a nonsingular action of a group G with MSJ, every quotient comes from a closed subgroup of the center of G whose action is totally non-ergodic. Thus, totally ergodic nonsingular flows with MSJ are prime. We then show an analogous result to Veech's theorem, namely that for a nonsingular action of a group G with GSJ, every quotient comes from a locally compact subgroup of the centralizer whose action is totally non-ergodic.