On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups

TL;DR

This paper establishes a duality relationship between the Cayley graph and the coset geometry of von Dyck groups, revealing a new structural connection in group theory.

Contribution

It introduces a vertex-to-edge duality linking the Cayley graph and coset geometry of von Dyck groups, a novel insight in geometric group theory.

Findings

Proves the duality between Cayley graph and coset geometry.

Provides a new perspective on the structure of von Dyck groups.

Enhances understanding of geometric representations of groups.

Abstract

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality.