HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H ⊆ 2 ω 1 that is an HFD with the following property(P) ojection of H onto every partial product 2 I for I ∈ [ ω 1 ] ω is onto. uch group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on 2 κ , there is an HFD topological group in 2 ω 1 which has property (P).