Pushing fillings in right-angled Artin groups. (Q2840160)

A polyhedral complex \(X\) is defined to be a CW-complex in which each cell is isometric to a convex polyhedron in Euclidean space and the gluing maps are isometries.NEWLINENEWLINE In a \(k\)-connected polyhedral complex is defined the \(k\)-th order Dehn function \(\delta_X^{k;cell}(\ell)\). If a group \(G\) acts geometrically on \(X\) then the growth rate of \(\delta_X^{k;cell}(\ell)\) depends only on \(G\), so \(\delta_G^{k;cell}(\ell)\) can be defined (see details in paragraph 2.1 in the paper).NEWLINENEWLINE Let \(X\) be a space, the divergence dimension \(\text{divdim}(X)\) is defined to be the largest integer \(k\) such that \(X\) is \((\rho,k)\)-acyclic at infinity for some \(0<\rho\leq 1\). For \(k\leq\text{divdim}(X)\) the divergence invariant \(\text{Div}^k(X)\) is defined as a two parameter family of functions. A partial order \(\preceq\) is defined for two divergence invariants \(\text{Div}^{k_1}(X)\) and \(\text{Div}^{k_2}(Y)\) and \(\text{Div}^{k_1}(X)\), \(\text{Div}^{k_2}(Y)\) are equivalent (\(\text{Div}^{k_1}(X)\asymp\text{Div}^{k_2}(Y)\)) if and only if \(\text{Div}^{k_1}(X)\preceq\text{Div}^{k_2}(Y)\) and \(\text{Div}^{k_1}(X)\succeq\text{Div}^{k_2}(Y)\) (see details in paragraph 2.2 in the paper).NEWLINENEWLINE A first result is the following: Proposition. Let \(X\) and \(Y\) be \(k\)-connected cell complexes with finitely many isometry types of cells. If \(X\) is quasi-isometric to \(Y\) and \(Y\) is \(k\)-acyclic at infinity, then \(\text{Div}^k(X)\asymp\text{Div}^k(Y)\).NEWLINENEWLINE This quasi-isometry invariance allows to write \(\text{Div}^k(G)\), instead of \(\text{Div}^k(X)\), if \(X\) has a geometric \(G\)-action.NEWLINENEWLINE Let \(\Gamma\) be a finite graph with no loops and multiple edges. If the vertices of \(\Gamma\) are labeled by \(a_1,a_2,\ldots,a_n\), then the right-angled Artin group defined over \(\Gamma\) is the group with the presentation \(A_\Gamma=\langle a_1,a_2,\ldots,a_n\mid [a_i,a_j]=1\) for each edge \(a_i,a_j\) of \(\Gamma\rangle\).NEWLINENEWLINE A very interesting subgroup of \(A_\Gamma\) is the subgroup \(H_\Gamma=\text{Ker }h\), where \(h\colon A_\Gamma\to\mathbb Z\) is the homomorphism which sends each generator to 1. The subgroup \(H_\Gamma\) is studied by \textit{M. Bestvina} and \textit{N. Brady} [Invent. Math. 129, No. 3, 445-470 (1997; Zbl 0888.20021)].NEWLINENEWLINE It is known that the group \(A_\Gamma\) acts freely on a \(\text{CAT}(0)\) cube-complex \(X_\Gamma\). The main purpose of the paper is to establish bounds for the \(k\)-th order Dehn function \(\delta_{H_\Gamma}^{k;cell}(\ell)\) and for the divergence invariant \(\text{Div}^k(A_\Gamma)\).NEWLINENEWLINE Theorem A. If the Bestvina-Brady group \(H_\Gamma\) is of type \(F_{k+1}\), then \(\delta_{H_\Gamma}^{k;cell}(\ell)\preceq\ell^{2(k+1)/k}\).NEWLINENEWLINE This theorem recovers a theorem of \textit{W. Dison} [in Bull. Lond. Math. Soc. 40, No. 3, 384-394 (2008; Zbl 1188.20038)].NEWLINENEWLINE In the paper is given the definition (Definition 5.3) of an orthoplex group.NEWLINENEWLINE Theorem B. If \(A_\Gamma\) is a \(k\)-orthoplex group, then \(\delta_{H_\Gamma}^{k;cell}(\ell)\succeq\ell^{2(k+1)/k}\).NEWLINENEWLINE Theorem C. For \(0\leq k\leq\text{divdim}(A_\Gamma)\), \(r^{k+1}\preceq\text{Div}^k(A_\Gamma)\preceq r^{2k+2}\).NEWLINENEWLINE The upper and lower bounds are sharp: for every \(k\) there are examples of right-angled Artin groups realizing these bounds.