Examples of geodesic ghor algebras on hyperbolic surfaces

TL;DR

This paper introduces geodesic ghor algebras on hyperbolic surfaces, extending properties of cancellative dimer algebras from tori to higher genus surfaces, and explores their central geometry.

Contribution

It defines a new class of quiver algebras on surfaces that retain desirable properties beyond the torus, linking algebraic and topological features.

Findings

Existence of nontrivial geodesic ghor algebras

Explicit descriptions of their central geometry

Extension of properties from tori to hyperbolic surfaces

Abstract

Cancellative dimer algebras on a torus have many nice algebraic and homological properties. However, these nice properties disappear for dimer algebras on higher genus surfaces. We consider a new class of quiver algebras on surfaces, called 'geodesic ghor algebras', that reduce to cancellative dimer algebras on a torus, yet continue to have nice properties on higher genus surfaces. These algebras exhibit a rich interplay between their central geometry and the topology of the surface. We show that (nontrivial) geodesic ghor algebras do in fact exist, and give explicit descriptions of their central geometry. This article serves a companion to the article 'A generalization of cancellative dimer algebras to hyperbolic surfaces', where the main statement is proven.