Locally compact strictly convex metric groups are abelian
Taras Banakh, Oles Mazurenko
TL;DR
The paper proves that locally compact strictly convex metric groups are abelian. It constructs a real multiplication on the group by embedding the real line via geodesics and shows that certain dyadic subgroups form a Z[1/2]-module, yielding a genuine real-parameter subgroup. Using these structures, it is shown that all compact subgroups must be trivial, and Iwasawa's theorem is applied to deduce abelianness of the whole group; as a consequence, such groups are finite-dimensional real normed spaces. This work tightly links metric convexity with classical group structure, constraining the geometry of locally compact strictly convex metric groups and highlighting a path from metric properties to abelian structure.
Abstract
We show that every locally compact strictly convex metric group is abelian, thus answering one problem posed by the authors in their earlir paper. To prove this theorem we first construct the isomorphic embeddings of the real line into the strictly convex metric group using its geodesic properties and charaterization of the real line as a unique not monothetic one-parametric metrizable topological group. We proceed to show that all compact subgroups in a strictly convex metric group are trivial, which combined with the classical result of Iwasawa completes the proof of the main result.