One-parameter subgroups of topological abelian groups

TL;DR

This paper characterizes the arc component of certain topological abelian groups as the union of one-parameter subgroups and shows the exponential map's openness in specific reflexive groups.

Contribution

It establishes a precise description of the arc component in a broad class of topological abelian groups and proves the openness of the exponential map in metrizable separable locally arc-connected reflexive groups.

Findings

Arc component equals union of one-parameter subgroups in certain groups

Exponential map is open in specific reflexive groups

Characterization applies to locally quasi-convex groups with onto evaluation map

Abstract

It is proved that, for a wide class of topological abelian groups (locally quasi--convex groups for which the canonical evaluation from the group into its Pontryagin bidual group is onto) the arc component of the group is exactly the union of the one--parameter subgroups. We also prove that for metrizable separable locally arc-connected reflexive groups, the exponential map from the Lie algebra into the group is open.