In this paper, we study the Borel structure of the space of left-orderings (Formula presented.) of a group (Formula presented.) modulo the natural conjugacy action, and by using tools from descriptive set theory, we find many examples of countable left-orderable groups such that the quotient space (Formula presented.) is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of (Formula presented.) on (Formula presented.) is universal, and leverage this result to provide many other examples of countable left-orderable groups (Formula presented.) such that the natural (Formula presented.) -action on (Formula presented.) induces a universal countable Borel equivalence relation.
Borel structures on the space of left-orderings
Calderoni F.;
2022-01-01
Abstract
In this paper, we study the Borel structure of the space of left-orderings (Formula presented.) of a group (Formula presented.) modulo the natural conjugacy action, and by using tools from descriptive set theory, we find many examples of countable left-orderable groups such that the quotient space (Formula presented.) is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of (Formula presented.) on (Formula presented.) is universal, and leverage this result to provide many other examples of countable left-orderable groups (Formula presented.) such that the natural (Formula presented.) -action on (Formula presented.) induces a universal countable Borel equivalence relation.
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