# Dimension and Borel reducibility

CUNY Logic Workshop, New York, April 2009

*Abstract*: Borel reducibility is a tool from descriptive set theory which can be used to measure the complexity of classification problems. Our starting point is the amazing result, due to Hjorth and Thomas, that the classification problem for torsion-free abelian groups of finite rank increases strictly in complexity with the rank. Other invariants besides just the rank can be used. For instance, Thomas showed that even once the rank is fixed, the classification subproblems for $p$-local and $q$-local groups have incomparable complexities. In each of these results, the “dimension” of the classification problem plays a crucial role. This left open the natural question: to what extent do the dimensions of two classification problems decide their relationship under Borel reducibility? In this talk, we will motivate and discuss all of the above notions, before giving a partial answer to this question.