Asymptotic dimension of graphs of groups and one-relator groups (Q6091924)

The \textit{asymptotic dimension} is an invariant of the large-scale geometry of a metric space, introduced by Gromov. Considering finitely generated groups, it thus serves a group invariant. The paper under review deals with the computation of this invariant for in certain important cases, mainly for one-relator groups, right-angled Artin groups and fundamental groups of finite graphs of groups. These results build on a bound on the asymptotic dimension of HNN extension. The author demonstrates that \(\operatorname{asdim}G\ast_N \leq \max\{\operatorname{asdim}G, \operatorname{asdim}N + 1\}\), analysing the interplay between the Cayley graph of the HNN extension and its Bass-Serre tree. Combining this result with a description of the asymptotic dimension of free products due to [\textit{G. C. Bell} et al., Fundam. Math. 183, No. 1, 39--45 (2004; Zbl 1068.20044)], a result of [\textit{N. Wright}, Geom. Topol. 16, No. 1, 527--554 (2012; Zbl 1327.20047)] on the asymptotic dimension of right-angled Artin groups is recovered. Next, the author demonstrates that the asymptotic dimension of a one-relator groups is bound from above by \(2\), thus proving a conjecture of Dranishnikov. While it was previously known that the dimension was finite, it is notable that this constitutes the first bound independent of length of the relator. Furthermore, the value of \(2\) is optimal. The proof of this result is uses an inductive argument on the length of the relator and the previously established bound concerning HHN extensions, which is crucial to obtain a uniform bound. Going further, the author completes his analysis by characterising the one-relator groups with asymptotic dimension \(0\) or \(1\). Finally, using similar methods, the author establishes an upper bound on the asymptotic dimension of fundamental groups of finite graphs of groups in terms of the dimension of their vertex and edge groups.