RESEARCH ON THE BOREL REDUCIBILITY AMONG EQUIVALENCE RELATIONS

Abstract

We recently found a kind of orbit equivalence relations, denoted by

, which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G , let

E ( G ) $E(G)$ upper E left parenthesis upper G right parenthesis

be the right coset equivalence relation

G ω / c ( G ) $G^\omega /c(G)$ upper G Superscript omega Baseline divided by c left parenthesis upper G right parenthesis

, where

c ( G ) $c(G)$ c left parenthesis upper G right parenthesis

is the group of all convergent sequences in G . We list some main results.

Let G and H be two Polish groups. (1) G is a discrete countable group containing at least two elements iff

; (2) G is non-archimedean iff

E ( G ) ≤ B = + $E(G)\le _B=^+$ upper E left parenthesis upper G right parenthesis less than or equals Subscript upper B Baseline equals Superscript plus Baseline

; (3) if H is CLI but G is not, then

E ( G ) ≰ B E H Y $E(G)\not \le _BE_H^Y$ upper E left parenthesis upper G right parenthesis neither less than nor equals Subscript upper B Baseline upper E Subscript upper H Superscript upper Y

for any Polish H -space Y ; (4) if H is a non-archimedean Polish group but G is not, then

E ( G ) ≰ B E H Y $E(G)\not \le _BE_H^Y$ upper E left parenthesis upper G right parenthesis neither less than nor equals Subscript upper B Baseline upper E Subscript upper H Superscript upper Y

for any Polish H -space Y ; (5) if H is TSI but G is not, then

E ( G ) ≰ B E ( H ) $E(G)\not \le _BE(H)$ upper E left parenthesis upper G right parenthesis neither less than nor equals Subscript upper B Baseline upper E left parenthesis upper H right parenthesis

; (6) G is TSI non-archimedean iff

E ( G ) ≤ B E 0 ω $E(G)\leq _B E_0^\omega $ upper E left parenthesis upper G right parenthesis less than or equals Subscript upper B Baseline upper E 0 Superscript omega

iff

E ( G ) ≤ B R ω / c 0 $E(G)\leq _B {\mathbb {R}}^\omega /c_0$ upper E left parenthesis upper G right parenthesis less than or equals Subscript upper B Baseline double struck upper R Superscript omega Baseline divided by c 0

; in particular,

E ( G ) ∼ B E 0 ω $E(G)\sim _B E_0^\omega $ upper E left parenthesis upper G right parenthesis tilde Subscript upper B Baseline upper E 0 Superscript omega

iff G is TSI uncountable non-archimedean.

The notion of

-l.m.-unbalanced Polish group for

α < ω 1 $\alpha <\omega _1$ alpha less than omega 1

is introduced. Let

G , H $G,H$ upper G comma upper H

be Polish groups,

0 < α < ω 1 $0<\alpha <\omega _1$ 0 less than alpha less than omega 1

. If G is

α $\alpha $ alpha

-l.m.-unbalanced but H is not, then

E ( G ) ≰ B E ( H ) $E(G)\not \le _B E(H)$ upper E left parenthesis upper G right parenthesis neither less than nor equals Subscript upper B Baseline upper E left parenthesis upper H right parenthesis

.

For TSI Polish groups, we proved a Pre-rigid Theorem: the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, we proved several rigid theorems for locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. In particular, we obtained a complete answer for abelian connected Lie groups: for

,

E ( R b 0 × T e 0 ) ≤ B E ( R b 1 × T e 1 ) $E({\mathbb {R}}^{b_0}\times {\mathbb {T}}^{e_0})\le _BE({\mathbb {R}}^{b_1}\times {\mathbb {T}}^{e_1})$ upper E left parenthesis double struck upper R Superscript b 0 Baseline times double struck upper T Superscript e 0 Baseline right parenthesis less than or equals Subscript upper B Baseline upper E left parenthesis double struck upper R Superscript b 1 Baseline times double struck upper T Superscript e 1 Baseline right parenthesis

iff

e 0 ≤ e 1 $e_0\le e_1$ e 0 less than or equals e 1

and

b 0 + e 0 ≤ b 1 + e 1 $b_0+e_0\le b_1+e_1$ b 0 plus e 0 less than or equals b 1 plus e 1

. In addition, we also proved that, the partially ordered set

P ( ω ) / Fin $P(\omega )/\text {Fin}$ upper P left parenthesis omega right parenthesis divided by Fin

can be embedded into Borel equivalence relations between

E ( R n ) $E({\mathbb {R}}^n)$ upper E left parenthesis double struck upper R Superscript n Baseline right parenthesis

and

E ( T n ) $E({\mathbb {T}}^n)$ upper E left parenthesis double struck upper T Superscript n Baseline right parenthesis

.