Super-sequences in the arc component of a compact connected group

TL;DR

This paper characterizes connected infinite compact abelian groups through the existence of special super-sequences in their arc component, linking topological and algebraic properties and extending previous theorems.

Contribution

It establishes a new characterization of connectedness in compact abelian groups using super-sequences and qc-density, and generalizes a known theorem by Aussenhofer.

Findings

Connected groups contain super-sequences converging to identity that generate dense subgroups.

Super-sequences in the arc component characterize the connectedness of the group.

The restriction homomorphism on characters is a topological isomorphism for connected locally compact abelian groups.

Abstract

Let G be an abelian topological group. The symbol \hat{G} denotes the group of all continuous characters \chi : G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that \chi(E) \subseteq \phi([-1/4,1/4]) holds only for the trivial character \chi \in \hat{G}, where \phi : R --> T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Aussenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : \hat{G} --> \hat{G}_a defined by r(\chi) = \chi\restriction_{G_a} for \chi \in \hat{G}, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component G_a contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \setminus {e} is an infinite suitable set for G in the sense of Hofmann and Morris).