  Cancellative dimer algebras on a torus are noncommutative crepant
resolutions, and in particular have many nice algebraic and homological
properties. All of these properties disappear, however, 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 possess nice properties on higher genus surfaces. We
show that noetherian localizations of these algebras are endomorphism rings of
modules over their centers, and establish a rich interplay between their
central geometry and the topology of the surface in which they are embedded.
