Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.


I think it will be a [Triangle group](https://en.wikipedia.org/wiki/Triangle_group), but I'm not sure.


(In spherical geometry, the answer is that there is no smallest tile, because you can make [bipyramids](https://en.wikipedia.org/wiki/Bipyramid) with arbitrarily small faces. I don't think this will be the case with the hyperbolic plane though.)



Smallest tile to tessellate the hyperbolic plane
The tilings mentioned by Ian Agol are related to an action of a Baumslag-Solitar group $\{ a,b \bigg| b^{-1}a^2b=a \}$ on the hyperbolic plane. They have arbitrarily small area, but diameter uniformly bounded away from $0$. It is possible to tesselate the hyperbolic plane with a single tile with arbitrarily small diameter, too. Let there be $n$ arcs on top and $n+1$ arcs on the bottom. As $n \to \infty$ the distance between the top and bottom goes to $0$. The region is [naturally related to a (non-faithful) action of a Baumslag-Solitar group](https://mathoverflow.net/questions/18411/do-the-baumslag-solitar-groups-occur-in-nature/18417#18417) $\{a,b \bigg|b^{-1}a^{n+1}b=a^n\}$ on the hyperbolic plane.


