An obstruction to embedding right-angled Artin groups in mapping class groups (Q2929673)

Let \(\Gamma\) be a finite simplicial graph; a right-angled Artin group (RAAG) is a group \(A(\Gamma)= \{V(\Gamma) : [u,v]=1 \text{ whenever } (u,v) \in E(\Gamma)\}\), where \(V (\Gamma)\) and \(E(\Gamma)\) are the set of vertices and edges of the structure \(\Gamma\) respectively. RAAG's have drawn attention in recent years because they have intrinsic subgroup richness and also because of their relations with cube complexes and their applications.NEWLINENEWLINEIn virtue of this, the main result [Theorem 1.2] of this work has its origins in the question: given a RAAG, \(A(\Gamma)\), how simple can a surface \(S\) be (what is the smallest value of the Euler characteristic \(|\chi (S)|\)), such that there is an embedding of the RAAG of \(\Gamma\) into the mapping class group (MCG) of the surface \(S\), \(A(\Gamma) \rightarrow \text{Mod} (S)\)? A question which is motivated by a preprint by \textit{J. Crisp} and \textit{B. Farb} [``The prevalence of surface groups in mapping class groups'', Preprint], by a single question of \textit{B. Farb} (ed.) [Problems on mapping class groups and related topics. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1098.58001)] and also by previous work by the authors. Contrary to the statement of the question the authors provide an obstruction motivated by restrictions on the surface given by \textit{J. S. Birman} et al. [Duke Math. J. 50, 1107--1120 (1983; Zbl 0551.57004)], that the maximal rank of an abelian subgroup in \(\text{Mod} (S)\) is equal to the maximal number of disjoint curves on \(S\), this latter proportional to \(|\chi (S)|\). So, if \(A(\Gamma)\) embeds in \(\text{Mod}(S)\), then the cohomological dimension of \(A(\Gamma)\) is at most equal to the maximal rank of an abelian subgroup of \(\text{Mod}(S)\).NEWLINENEWLINEThe main goal of the work, as stated, is to provide an obstruction from comparing chromatic numbers of the complex \(\Gamma\) and the graph of the complex of curves of some surface. More precisely, given a surface \(S\) with \(0<|\chi (S)|< \infty\), they find an a finite graph, \(\Gamma\), of girth \(\geq M>0\), such that the RAAG of \(\Gamma\) does not embed in \(\text{Mod}(S)\). Assuming that \(A(\Gamma) < \text{Mod}(S)\), then \(\Gamma\) embeds in \(\mathcal{C}(S)_k\), as a subgraph, by Lemma 2.3. And by a theorem of Bestvina-Bromberg-Fujiwara, see [\textit{M. Bestvina} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 1--64 (2015; Zbl 1372.20029)], \(\mathcal{C}(S)_k\) has finite chromatic number, \(N\), which cannot be exceeded by the chromatic number of \(\Gamma\). Finally, by a theorem of Erdős, there exists a graph \(\Gamma\) of girth \(\geq M >0\) and chromatic number \(>N\). One of the possible applications of these techniques can lead to the future study of the (multi-)curve complexes, or even the topology of 3-manifolds from another perspective.