On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

Abstract

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some class of finite p-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J. D. Hamkins, and R. Miller (2012). A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.