Geodesic
On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $\mathfrak{X}(G)$
Dessislava H. Kochloukova1
1State University of Campinas (UNICAMP), Brazil
Groups, geometry, and dynamics, Tome 17 (2023) no. 1, pp. 215-243

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DOI

For a finitely generated group G, we calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(X(G)) for the weak commutativity construction X(G). Identifying S(X(G)) with S(X(G)/W(G)), we show Σ2(X(G),Z)⊆Σ2(X(G)/W(G),Z) and Σ2(X(G))⊆Σ2(X(G)/W(G)), that are equalities when W(G) is finitely generated, and we explicitly calculate Σ2(X(G)/W(G),Z) and Σ2(X(G)/W(G)) in terms of the Σ-invariants of G. We calculate completely the Σ-invariants in dimensions 1 and 2 of the group ν(G) and show that if G is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square G⊗G is finitely presented.
DOI : 10.4171/ggd/680
Classification : 20-XX
Mots-clés : Homological and homotopical finiteness properties, Σ-invariants, weak commutativity

Dessislava H. Kochloukova  1

1 State University of Campinas (UNICAMP), Brazil 
Dessislava H. Kochloukova. On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $\mathfrak{X}(G)$. Groups, geometry, and dynamics, Tome 17 (2023) no. 1, pp. 215-243. doi: 10.4171/ggd/680
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     author = {Dessislava H. Kochloukova},
     title = {On the {Bieri{\textendash}Neumann{\textendash}Strebel{\textendash}Renz} invariants of the weak commutativity construction $\mathfrak{X}(G)$},
     journal = {Groups, geometry, and dynamics},
     pages = {215--243},
     year = {2023},
     volume = {17},
     number = {1},
     doi = {10.4171/ggd/680},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/680/}
}
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