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Register a new userOn the Bieri-Neumann-Strebel-Renz invariants and limit groups over Droms RAAGs
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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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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 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.
We calculate the Bieri-Neumann-Strebel-Renz invariant $Sigma^1(G)$ for even Artin groups $G$ with underlying graph $Gamma$ such that if there is a closed reduced path in $Gamma$ with all labels bigger than 2 then the length of such path is always odd. We show that $Sigma^1(G)^c$ is a rationally defined spherical polyhedron.
We generalize some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if $S$ is a full subdirect product of type $FP_s(mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of $S$ to the direct product of any $s$ of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to $m$. In particular, this gives the values of the analytic $L^2$-Betti numbers of these groups in the respective dimensions.
We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.
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