I like the hyperbolic plane where Escher's "circle limits" live.
In the hyperbolic plane you can turn your car (i.e. constant aceleration perpendicular to the constant speed) and not manage to close its trajectory (there are equidistant curves and horocycles).
Also, the symmetry group of the tiling by right angled hexagons contains all but a finite number of closed surface groups. So you can build almost all closed surfaces gluing these hexagons.
People in the hyperbolic plane wont agree on the angle between two stars (i.e. boundary points) but if you average the measurements of other people around you the result will agree with your own measurement (hence everyone thinks they are right).
The modular group (and its congruence subgroups) are important in number theory (which I know next to nothing about) and also in complex analysis (where, for example, the congruence subgroup $\Gamma(2)$ is the covering group of the plane minus two points and allows one to prove that if an entire function omits two values in its image it must be constant).
The list could go on (the Gauss-Bonnet theorem, Brownian motion escapes with positive speed to infinity, Anosov property the geodesic flow, quasi-geodesics are at bounded distance from geodesics, the area of a convex hull is bounded by a constant times the number of points, the strong isoperimetric inequality holds i.e. perimeter is greater then volume for all sets, etc, ...).