On the geometry of van Kampen diagrams of graph products of groups

In this article, we propose a geometric framework dedicated to the study of van Kampen diagrams of graph products of groups. As an application, we find information on the word and the conjugacy problems. The main new result of the paper deals with the computation of conjugacy length functions. More precisely, if $\Gamma$ is a finite graph and $\mathcal{G}= \{ G_u \mid u \in V(\Gamma) \}$ a collection of finitely generated groups indexed by the vertices of $\Gamma$, then $$\max\limits_{u \in V(\Gamma)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{\Gamma \mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{\Delta \subset \Gamma \ \text{complete}} \sum\limits_{u \in V(\Delta)} \mathrm{CLF}_{G_u}(n)$$ for every $n \geq 1$, where $D$ denotes the maximal diameter of a connected component of the opposite graph $\Gamma^{\mathrm{opp}}$. As a consequence, a graph product of groups with linear conjugacy length functions has linear conjugacy length function as well.