2023/05/08 by Bachir Bekka, Bekka, Bachir
Mathematics · #20E18 #22C05 #22D10 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2305.04803
openalex publication_date 2023/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G= N\rtimes H be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification \rm Bohr(G) and the profinite completion \rm Prof(G) of G are, respectively, isomorphic to semi-direct products Q1 \rtimes \rm Bohr(H) and Q2 \rtimes \rm Prof(H) for appropriate quotients Q1 of \rm Bohr(N) and Q2 of \rm Prof(N). We give a precise description of Q1 and Q2 in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have \rm Bohr(G)≅ A \rtimes \rm Bohr(H) and \rm Prof(G)≅ B \rtimes \rm Prof(H), where A is the group of unitary characters of N with finite H-orbits and B the subgroup of A of characters with finite image. Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where G= Λ\wr H is a wreath product of countable groups; we show in particular that \rm Bohr(Λ\wr H) is isomorphic to \rm Bohr(Λ\rm Ab\wr H) and \rm Prof(Λ\wr H) is isomorphic to \rm Prof(Λ\rm Ab \wr H), where Λ\rm Ab=Λ/ [Λ, Λ] is the abelianization of Λ. As examples, we compute \rm Bohr(G) and \rm Prof(G) when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.