New constructions of free products and geodetic Cayley graphs

2025-09-26

AI Summary

This work addresses the challenge of systematically constructing Cayley graphs with geodesic properties, overcoming limitations of existing free product representations that hinder the generation of novel geodesic graphs. Methodologically, we refine the graph subdivision technique of Parthasarathy and Srinivasan and, for the first time, lift geodesic preservation from the graph-theoretic level to the realm of group representations and string rewriting systems. This yields a novel rewriting system based on the free product group $G ast F_n$, associated with a newly constructed generating set $Sigma$. Our approach breaks classical constraints, producing infinitely many previously unknown families of geodesic Cayley graphs—substantially broadening the constructive paradigm for geodesic groups. The results provide a crucial construction tool for the geodesic group conjecture and foster interdisciplinary advancement at the intersection of group theory, combinatorial group theory, and geometric group theory.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Location- and context-aware Web and WoT applications and services

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 $Σ$ and an integer $n$, our construction produces a rewriting system presenting $G ast F_{n|Σ|}$ 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.

Problem

• Constructing new free product presentations with geodetic Cayley graphs
• Adapting subdivision techniques to group presentations and rewriting systems
• Producing infinite families of geodetic Cayley graphs for conjecture investigations

Innovation

• Systematic method constructs new free product presentations
• Adapts subdivision techniques to group rewriting systems
• Generates infinite families of geodetic Cayley graphs