New constructions of free products and geodetic Cayley graphs

TL;DR

The paper develops a subdivision-based method to construct new geodetic Cayley graphs by translating edge subdivisions of a geodetic Cayley graph into a rewriting-system framework. It defines nabla_n(G, Sigma), an inverse-closed confluent rewriting system whose presented group is G * F_{n|Sigma|}, and proves that geodeticity of the original generating set transfers to the expanded generating set of the free product. The results provide a general framework for generating infinite families of geodetic Cayley graphs and offer insights into longstanding conjectures about geodetic groups. While not resolving the major conjectures, the construction yields systematic, scalable ways to explore geodetic structures and their applications, including potential uses in parallel computing and further geometric group theory investigations.

Abstract

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.

Figures (6)

• Figure 1: Geodetic Cayley graphs for $F(X)$ discussed in \ref{['eg:shortcutinfreegroups']}.
• Figure 2: The subdivision graph of a $5$-cycle graph with $n=1$.
• Figure 3: Paths $\alpha,\beta,\gamma$ in the proof of \ref{['lem:uniformsubdivision']}.
• Figure 4: $\Gamma=\text{Cay}(G,\Sigma)$ with $\Sigma$-labelling $L$.
• Figure 5: $\Gamma(1)$ with $\Sigma(1)$-labelling $L(1)$.

Theorems & Definitions (39)

• Example 1.1
• Example 1.2
• Example 1.3
• Definition 2.1: Graph isomorphism
• Definition 2.2: Labelled graph isomorphism
• Definition 2.3: $\Gamma$-slex
• Lemma 2.4
• proof
• Lemma 2.5
• proof