A note on intrinsic topologies of groups

TL;DR

This paper explores various natural topologies on groups derived from their algebraic structures, revealing differences and coincidences among them, especially in abelian, algebraically free, and symmetric groups.

Contribution

It demonstrates the existence of a countable abelian group where bounded and full Zariski topologies differ, and shows the hyperconnectivity of the semigroup Zariski topology on algebraically free groups, also identifying the topology on symmetric groups.

Findings

Existence of a countable abelian group with distinct bounded and full Zariski topologies.

Semigroup Zariski topology is hyperconnected on algebraically free groups.

On symmetric groups, the semigroup Hausdorff-Markov topology matches pointwise convergence.

Abstract

We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that there exists a countable abelian group in which no bounded version of the Zariski topology coincides with the full Zariski topology. Complementing a recent result by Goffer and Greenfeld, we show that on any group with no algebraicity the semigroup Zariski topology is hyperconnected and hence, in many cases, is distinct from the group Zariski topology. Finally, we show that on the symmetric groups, the semigroup Hausdorff-Markov topology coincides with the topology of pointwise convergence.