For example, for categoricity, general problem is impossible; for example Downey and
Montalban showed that the isomorphism problem for torsion-free abelian groups is
Sigma_1^1-complete. The principle difficulty lies in the lack of invariants. However,
where there are some invariants there we can salvage some effectiveness. The groups
we look at are the completely decomposable ones, which have decompositions of the
form oplus_{i in omega} G_i with G_i a subgroup of the additive group of the rationals.
Such groups are called homogeneous if G_i=H for all i. Alexander Melnikov and the
author have shown that homogeneous computable completely decomposable groups are
always Delta_3^0 categorical, this bound is sharp, and have classified when the groups
are Delta_2^0 categorical in terms of what are called semilow sets. In more recent work,
we have shown that every computable completely decomposable group is Delta_5^0
categorical and that this bound is sharp. Additionally we can show that the index set
of such groups is Sigma_7^0. I will also describe ongoing work on Ulm's Theorem.
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