Cohomology and extensions of relative Rota-Baxter groups (Q6655050)

In relation to the Yang-Baxter equation, there is the notion of skew braces (introduced by \textit{L. Guarnieri} and \textit{L. Vendramin} [Math. Comput. 86, No. 307, 2519--2534 (2017; Zbl 1371.16037)] as a generalization of braces) and the notion of Rota-Baxter groups (introduced by \textit{L. Guo} et al. [Adv. Math. 387, Article ID 107834, 34 p. (2021; Zbl 1468.17026)] as an analogue of Rota-Baxter operators defined on algebras).\N\NThe reviewer reports the authors' abstract: ``Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation'' (see [\textit{V. G. Bardakov} and \textit{V. Gubarev}, J. Algebra 596, 328--351 (2022; Zbl 1492.17019)]). ``In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two.''