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Register a new userOn the Bieri-Neumann-Strebel-Renz $Sigma^1$-invariant of even Artin groups
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Authors
Dessislava H. Kochloukova
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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.
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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 $Sigma^n(H)$.
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 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.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all $2$-dimensional Artin groups, and for spherical Artin groups of any type other than $E_6$, $E_7$, $E_8$. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.
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