  This paper is concerned with elliptic operators on plane tessellations. We
show that such an operator does not admit a compactly supported eigenfunction,
if the combinatorial curvature of the tessellation is nonpositive. Furthermore,
we show that the only geometrically finite, repetitive plane tessellations with
nonpositive curvature are the regular $(3,6), (4,4)$ and $(6,3)$ tilings.
