On topologizable and non-topologizable groups

Abstract

A group $G$ is called hereditarily non-topologizable if, for every $H\le G$, no quotient of $H$ admits a non-discrete Hausdorff topology. We construct first examples of infinite hereditarily non-topologizable groups. This allows us to prove that c-compactness does not imply compactness for topological groups. We also answer several other open questions about c-compact groups asked by Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing topologizable groups based on generic properties in the space of marked $k$-generated groups. As an application, we show that there exist non-discrete quasi-cyclic groups of finite exponent; this answers a question of Morris and Obraztsov.

References in corpus (6)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Peripheral fillings of relatively hyperbolic groups
• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups
• Small cancellation in acylindrically hyperbolic groups
• Acylindrically hyperbolic groups
• Hereditarily non-topologizable groups

Cited by in corpus (7)

• Small cancellation in acylindrically hyperbolic groups
• Categorically closed topological groups
• Topologization of sets endowed with an action of a monoid
• Quasi-isometric diversity of marked groups
• Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones
• Hereditarily minimal topological groups
• Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups