The action dimension of right-angled Artin groups (Q2788660)

The action dimension of a discrete group \(\Gamma\), denoted \(\mathrm{actdim}\Gamma\), is the smallest integer \(n\) such that \(\Gamma\) acts properly on a contractible \(n\)-manifold. By a theorem of Stallings, it is known that \(\mathrm{actdim} \Gamma \leq 2 \mathrm{gd} \Gamma\), where \(\mathrm{gd} \Gamma\) denotes the geometric dimension of \(\Gamma\), the minimum dimension of a model for \(B\Gamma\).NEWLINENEWLINEIn this paper, the authors study the action dimension of a right-angled Artin group \(A_L\) associated to a flag complex \(L\). They prove the following theorem:NEWLINENEWLINE\noindent \textbf{Theorem.} Suppose that \(L\) is a \(k\)-dimensional flag complex.NEWLINENEWLINE(1) If \(H_k (L; \mathbb{Z}/2 ) \neq 0\), then \(\mathrm{actdim}A_L=2k+2=2 \mathrm{gd}\Gamma\).NEWLINENEWLINE(2) If \(H_k (L; \mathbb{Z}/2)=0\) and \(k\neq 2\), then \(\mathrm{actdim}A_L\leq 2k+1\).NEWLINENEWLINEWhen \(L\) is a \(k\)-dimensional complex, the \(\ell ^2\)-dimension of \(A_L\) satisfies the inequality \(\ell ^2 \dim A_L\leq k+1\). Hence, part (1) of the above theorem gives that \(\mathrm{actdim} A_L \geq 2 \ell ^2 \dim A_L\). This shows that the action dimension conjecture holds for \(A_L\) when \(L\) is a \(k\)-dimensional flag complex with \(H_k (L ; \mathbb{Z} /2)\neq 0\). Note that the action dimension conjecture, due to Davis and Okun, states that for a discrete group \(\Gamma\), \(\mathrm{actdim}\Gamma\geq 2 \ell^2\dim \Gamma\).NEWLINENEWLINETo prove the theorem, the authors consider the octahedralization \(OL\) of the complex \(L\). They show that NEWLINE\[NEWLINE\mathrm{embdim}OL +1 \geq \mathrm{actdim}A_L \geq \mathrm{vkdim} OL +2,NEWLINE\]NEWLINE where \(\mathrm{embdim}OL\) denotes the embedding dimension of \(OL\), and \( \mathrm{vkdim} OL \) denotes the van Kampen dimension of \(OL\) (both defined in the introduction of the paper). To prove the first part of the theorem, they show that if \(H_k (L; \mathbb{Z}/2)\neq 0\), then \(\mathrm{vkdim} OL=2k\). For the second statement, they prove that if \(H_k (L; \mathbb{Z}/2)=0\) and \(k\neq 2\), then \(\mathrm{embdim} OL \leq 2k\).