November 15
Russell Miller,
CUNY
Computable reductions on groups and fields
Hjorth and Thomas established that the complexity of the isomorphism problem for torsion-free abelian groups of finite rank grows dramatically higher as the rank increases: for each $r$, there is no Borel function $F$ that maps each rank-$(r+1)$ group $G$ to a rank-$r$ group $F(G)$ in such a way that $G_0\cong G_1\iff F(G_0)\cong F(G_1)$. We say that there is no Borel reduction from isomorphism on $\operatorname{TFAb}_{r+1}$ to isomorphism on $\operatorname{TFAb}_r$. (From lower to higher rank, in contrast, such a reduction is readily seen.) Fields of transcendence degree $r$ over $\mathbb Q$ have very similar computability properties to groups in $\operatorname{TFAb}_r$. This being so, we extend their investigations to include the isomorphism relations on the classes $\operatorname{FD}_r$ of such fields. We show that there do exist reductions (not merely Borel, but actually computable, and moreover functorial) from each $\operatorname{TFAb}_r$ to the corresponding $\operatorname{FD}_r$, and also from each $\operatorname{FD}_r$ to $\operatorname{FD}_{r+1}$ (which proves more challenging than it was for the groups!). It remains open whether a theorem analogous to that of Hjorth-Thomas holds for the fields, but we use the notion of countable reductions to show that the fundamental obstacle to a reduction from $\operatorname{TFAb}_{r+1}$ to $\operatorname{TFAb}_r$ is the uncountability of these spaces. This is joint work with Meng-Che 'Turbo' Ho and Julia Knight.