New constructions of free products and geodetic Cayley graphs

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.