We give an algebraic characterization of those sequences
(
H
n
)
({H_n})
of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of
H
0
×
H
1
×
H
2
×
⋯
{H_0} \times {H_1} \times {H_2} \times \cdots
are Borel. In particular, the equivalence relations induced by Borel actions of
H
ω
{H^\omega }
,
H
H
countable abelian, are Borel iff
H
≃
⊕
p
(
F
p
×
Z
(
p
∞
)
n
p
)
H \simeq { \oplus _p}({F_p} \times \mathbb {Z}{({p^\infty })^{{n_p}}})
, where
F
p
{F_p}
is a finite
p
p
-group,
Z
(
p
∞
)
\mathbb {Z}({p^\infty })
is the quasicyclic
p
p
-group,
n
p
∈
ω
{n_p} \in \omega
, and
p
p
varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.