How complicated are countable torsion-free abelian groups? In particular, are they as complicated as countable graphs? In recent joint work with Shelah, we show it is consistent with ZFC that countable torsion-free abelian groups are $a \Delta^1_2$ complete; in other words, countable graphs can be encoded into them via an absolutely $\Delta^1_2$-map. I discuss this, and the related result: assuming large cardinals, it is independent of ZFC if there is an absolutely $\Delta^1_2$ reduction from Graphs to Colored Trees, which takes non-isomorphic graphs to non-biembeddable colored trees.