RESEARCH ON THE BOREL REDUCIBILITY AMONG EQUIVALENCE RELATIONS E ( G ) $E(G)$ upper E left parenthesis upper G right parenthesis

We recently found a kind of orbit equivalence relations, denoted by E ( G ) $E(G)$ upper E left parenthesis upper G right parenthesis , 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^ω /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 E ( G ) ∼ B E 0 E(G)~ BE₀ upper E left parenthesis upper G right parenthesis tilde Subscript upper B Baseline upper E 0 ; (2) G is non-archimedean iff E ( G ) ≤ B = + E(G)≤ 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