Non-topologizable groups
by
Gábor Lukács
Dalhousie University

Following Ol'shanskii, an infinite group G is non-topologizable if the only Hausdorff group topology on G is the discrete one. Given a monomial f(x)=g₀x^k₁g₂ x^k₂g₃...g_n-1 x^k_n g_n in a single variable x, with g_i ∈ G and k_i ∈ Z, the set V(f) = {g ∈ G | f(g)=e } is a closed subset in any Hausdorff group topology on G. Thus, if there are monomials f₁, ..., f_n such that G\{e}=V(f₁) ∪...∪V(f_n), then {e} is open in any Hausdorff group topology on G, and therefore G is non-topologizable. It turns out that for countable groups, the converse is also true. (A nice proof of this fact is available in "Topologies on groups determined by sequences" by Zelenyuk and Protasov.) Non-topologizable groups must have a very strongly non-commutative structure. For instance, if an abelian group A is non-topologizable, then it must be finite.

In this talk, we discuss the topological group theoretical aspects of the following problem:

Problem. Is there an infinite group G such that for every subgroup H ≤ G and every normal subgroup N of H, H/N is non-topologizable?

Date received: February 16, 2006