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Register a new userOn the Bieri-Neumann-Strebel-Renz invariants of the weak commutativity construction $X(G)$
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Authors
Dessislava H. Kochloukova
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For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $Sigma^1(X(G))$ for the weak commutativity construction $X(G)$. Identifying $S(X(G))$ with $S(X(G) / W(G))$ we show $Sigma^2(X(G),Z) subseteq Sigma^2(X(G)/ W(G),Z)$ and $Sigma^2(X(G)) subseteq $ $ Sigma^2(X(G)/ W(G))$ that are equalities when $W(G)$ is finitely generated and we explicitly calculate $Sigma^2(X(G)/ W(G),Z)$ and $ Sigma^2(X(G)/ W(G))$ in terms of the $Sigma$-invariants of $G$. We calculate completely the $Sigma$-invariants in dimensions 1 and 2 of the group $ u(G)$ and show that if $G$ is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square $G otimes G$ is finitely presented.
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We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups $H$ and $K$ of type $FP_n$ such that $[H,H] subseteq K subseteq H$ and a character $chi : K to mathbb{R}$ with $chi([H,H]) = 0$ we have $[chi] in Sigma^n(K, mathbb{Z})$ if and only if $[mu] in Sigma^n(H, mathbb{Z})$ for every character $mu : H to mathbb{R}$ that extends $chi$. The same holds for the homotopical invariants $Sigma^n(-)$ when $K$ and $H$ are groups of type $F_n$. We use these criteria to complete the description of the $Sigma$-invariants of the Bieri-Stallings groups $G_m$ and more generally to describe the $Sigma$-invariants of the Bestvina-Brady groups. We also show that the only if direction of such criterion holds if we assume only that $K$ is a subnormal subgroup of $H$, where both groups are of type $FP_n$. We apply this last result to wreath products.
For a group $G$ that is a limit group over Droms RAAGs such that $G$ has trivial center, we show that $Sigma^1(G) = emptyset = Sigma^1(G, mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $Sigma^1(H)$ and $Sigma^2(H)_{dis}$. In addition, we obtain a necessary condition for $[chi]$ to belong to $Sigma^n(H)$.
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