  We study a new class of quiver algebras on surfaces, called 'geodesic ghor
algebras'. These algebras generalize cancellative dimer algebras on a torus to
higher genus surfaces, where the relations come from perfect matchings rather
than a potential. Although cancellative dimer algebras on a torus are
noncommutative crepant resolutions, the center of any dimer algebra on a higher
genus surface is just the polynomial ring in one variable, and so the center
and surface are unrelated. In contrast, we establish a rich interplay between
the central geometry of geodesic ghor algebras and the topology of the surface
in which they are embedded. Furthermore, we show that noetherian central
localizations of such algebras are endomorphism rings of modules over their
centers.
