We develop a correspondence between Borel equivalence relations induced by closed subgroups of
S
∞
S_\infty
and weak choice principles, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
For example, we show that the equivalence relation
≅
ω
+
1
,
0
∗
\cong ^\ast _{\omega +1,0}
is strictly below
≅
ω
+
1
,
>
ω
∗
\cong ^\ast _{\omega +1,>\omega }
in Borel reducibility. By results of Hjorth-Kechris-Louveau,
≅
ω
+
1
,
>
ω
∗
\cong ^\ast _{\omega +1,>\omega }
provides invariants for
Σ
ω
+
1
0
\Sigma ^0_{\omega +1}
equivalence relations induced by actions of
S
∞
S_\infty
, while
≅
ω
+
1
,
0
∗
\cong ^\ast _{\omega +1,0}
provides invariants for
Σ
ω
+
1
0
\Sigma ^0_{\omega +1}
equivalence relations induced by actions of abelian closed subgroups of
S
∞
S_\infty
.
We further apply these techniques to study the Friedman-Stanley jumps. For example, we find an equivalence relation
F
F
, Borel bireducible with
=
+
+
=^{++}
, so that
F
↾
C
F\restriction C
is not Borel reducible to
=
+
=^{+}
for any non-meager set
C
C
. This answers a question of Zapletal, arising from the results of Kanovei-Sabok-Zapletal (2013).
For these proofs we analyze the symmetric models
M
n
M_n
,
n
>
ω
n>\omega
, developed by Monro (1973), and extend the construction past
ω
\omega
, through all countable ordinals. This answers a question of Karagila (2019).