# Invariant Descriptive Set Theory

@inproceedings{Gao2008InvariantDS, title={Invariant Descriptive Set Theory}, author={Su Gao}, year={2008} }

Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets Coanalytic sets and pi 1/1 sets The Gandy-Harrington topology Polish Groups Metrics on topological groups Polish groups Continuity of homomorphisms The permutation group S Universal Polish groups The Graev metric groups Polish Group Actions Polish G-spaces The Vaught transforms Borel G-spaces Orbit…

## 325 Citations

Polish groupoids and functorial complexity

- Mathematics
- 2014

We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring the complexity of classifying the objects of a category in a constructive and…

Borel equivalence relations induced by actions of tsi Polish groups

- Mathematics
- 2021

We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable…

DICHOTOMY FOR TSI POLISH GROUPS I: CLASSIFICATION BY COUNTABLE STRUCTURES

- Mathematics
- 2020

We introduce a property of orbit equivalence relation that we call property (IC) and show that a Borel orbit equivalence relation E G induced by a continuous action of a tsi Polish group G on a…

FORCING CONSTRUCTIONS AND COUNTABLE BOREL EQUIVALENCE RELATIONS

- MathematicsThe Journal of Symbolic Logic
- 2022

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hid- den regularity properties of Borel complete sections on…

Graev Metrics and Isometry Groups of Polish Ultrametric Spaces

- Mathematics
- 2013

This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces. In Chapter 2,…

DICHOTOMY FOR TSI POLISH GROUPS II: E3-DICHOTOMY

- Mathematics
- 2020

We investigate two properties of orbit equivalence relations, admitting a Gbounded topology and property (IC), and derive several consequences in the theory of Borel reducibility of Borel orbit…

The effective theory of Borel equivalence relations

- MathematicsAnn. Pure Appl. Log.
- 2010

Abelian pro‐countable groups and non‐Borel orbit equivalence relations

- MathematicsMath. Log. Q.
- 2016

T tame groups in this class are characterized, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$ are characterized.

LECTURE NOTES: AN INTRODUCTION TO COUNTABLE BOREL EQUIVALENCE RELATIONS AND ACTIONS OF COUNTABLE GROUPS

- Mathematics
- 2019

Borel measurable actions of countable groups have been intensively studied in the last decades from the viewpoints of descriptive set theory, dynamics and measured group theory [Gab10, Eps08, CM17,…

On Polish groups admitting non-essentially countable actions

- MathematicsErgodic Theory and Dynamical Systems
- 2020

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not…