A connected graph is called \emph{geodetic} if there is a unique shortest path between each pair of vertices. We introduce a systematic method for constructing new presentations of free products that give rise to previously unknown geodetic Cayley graphs. Our approach adapts subdivision techniques of Parthasarathy and Srinivasan (J. Combin. Theory Ser. B, 1982), which preserve geodecity at the graph level, to the setting of group presentations and rewriting systems. Specifically, given a group $G$ with geodetic Cayley graph with respect to generating set $Σ$ and an integer $n$, our construction produces a rewriting system presenting the free product of $G$ with a free group of rank $n|Σ|$ with geodetic Cayley graph with respect to a new generating set. This framework provides new infinite families of geodetic Cayley graphs and extends the toolkit for investigating long-standing conjectures on geodetic groups.

翻译：若连通图中任意一对顶点间存在唯一最短路径，则称该图为\\emph{测地图}。本文提出一种系统化方法，用于构造自由积的新表示，从而产生先前未知的测地凯莱图。我们的方法将Parthasarathy与Srinivasan（J. Combin. Theory Ser. B, 1982）在图形层面保持测地性的细分技术，适配至群表示与重写系统的框架中。具体而言，给定群$G$及其关于生成集$Σ$的测地凯莱图，以及整数$n$，我们的构造能生成一个重写系统，该系统表示$G$与秩为$n|Σ|$的自由群的自由积，并得到关于新生成集的测地凯莱图。此框架提供了新的无限族测地凯莱图，扩展了研究测地群中长期猜想的方法体系。