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Abstract

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 $\Sigma$ and an integer $n$, our construction produces a rewriting system presenting $G \ast F_{n|\Sigma|}$ with geodetic Cayley graph with respect to the 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.