Graev metric groups and Polishable subgroups

AbstractWe prove a conjecture of Hjorth: There is an uncountable Polish group all of whose abelian subgroups are discrete. We first construct directly a witness to Hjorth's conjecture. Then we consider an existing example in the literature. The example is the metric completion of a free topological group constructed by Graev. We give a definition slightly more general than Graev's and prove some properties of the Graev metrics which seem to be unknown previously. We also consider the problem of finding Polishable subgroups of the Graev metric groups with arbitrarily high Borel rank. In doing this we prove some general theorems on extensions of Polish groups with this property