Embedding of locally compact Hausdorff topological gyrogroups in topological groups

Extension of Haar’s theorem

“…As indicated in Example 1.1 of [14], the aforementioned Möbius gyrogroup can be embedded in the general linear group GL 2 (C) via the topological embedding a → 1 a ā 1 for all a ∈ D. This suggests studying connections between Haar measures on groups and on gyrogroups. We emphasize that an explicit formula for the Haar measure on the open disk D is known.…”

“…Let G be a locally compact Hausdorff topological gyrogroup. As shown in [14], the group A gyr generated by all the gyroautomorphisms of G,…”

“…for all a, b ∈ G and for all τ, σ ∈ A gyr . Furthermore, according to Theorem 3.9 of [14], G is embedded in Γ(G, A gyr ) via the topological embedding a → (a, i), where i is the identity function of G. By Theorem 3.5, if G is strong, then G possesses a unique Haar measure. However, the existence of a Haar measure on Γ(G, A gyr ) is in question because the local compactness of this group is not known.…”

“…This result extends Haar's theorem to the case of gyrogroups. As shown in [14], every locally compact Hausdorff topological gyrogroup G can be embedded in a completely regular topological group Γ(G, A gyr ). We then find a natural relationship between Haar measures on gyrogroups and on their related groups.…”

Embedding into connected, locally connected topological gyrogroups

“…Then Cai, Lin and He in [7] proved that every topological gyrogroup is a rectifiable space, which deduced that every first-countable topological gyrogroup is metrizable. Moreover, Wattanapan, Atiponrat and Suksumran [12] proved that every locally compact Hausdorff topological gyrogroup can be embedded in a completely regular topological group as a twisted subset. Since it is obvious that each topological group is a topological gyrogroup, it is natural to consider whether some well-known results of topological groups can be extended to topological gyrogroups.…”