关于群的图积的van Kampen图的几何
On the geometry of van Kampen diagrams of graph products of groups
摘要：
在这篇文章中，我们提出了一个几何框架，致力于研究的van Kampen图的图形产品的群体。作为一个应用程序，我们发现信息的字和共轭问题。本文的主要新结果涉及共轭长度函数的计算。更准确地说，如果$Γ$是一个有限图，而$\mathcal{G}= \{ G_u \mid u \in V(Γ) \}$是由$Γ$的顶点索引的有限生成群的集合，那么对于每个$n \geq 1$，都有$$\max\limits_{u \in V(Γ)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{Γ\mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{Δ\subset Γ\ \text{complete}} \sum\limits_{u \in V(Δ)} \mathrm{CLF}_{G_u}(n)$$，其中$D$表示相对图$Γ^{\mathrm{opp}}$的连通分支的最大直径。因此，具有线性共轭长度函数的群的图积也具有线性共轭长度函数。
英文摘要：
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 $Γ$ is a finite graph and $\mathcal{G}= \{ G_u \mid u \in V(Γ) \}$ a collection of finitely generated groups indexed by the vertices of $Γ$, then $$\max\limits_{u \in V(Γ)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{Γ\mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{Δ\subset Γ\ \text{complete}} \sum\limits_{u \in V(Δ)} \mathrm{CLF}_{G_u}(n)$$ for every $n \geq 1$, where $D$ denotes the maximal diameter of a connected component of the opposite graph $Γ^{\mathrm{opp}}$. As a consequence, a graph product of groups with linear conjugacy length functions has linear conjugacy length function as well.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
