EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES

We construct a non-abelian extension Γ of S1 by Z/3 × Z/3, and prove that Γ acts freely and smoothly on S5 × S5. This gives new actions on S5 × S5 for an infinite family 𝒫 of finite 3-groups. We also show that any finite odd-order subgroup of the exceptional Lie group G2 admits a free smooth action on S11 × S11. This gives new actions on S11 × S11 for an infinite family ℰ of finite groups. We explain the significance of these families 𝒫, ℰ for the general existence problem, and correct some mistakes in the literature.