The sum of the interior angles in geodesic and translation triangles of SL2(R) geometry

Abstract

We study the interior angle sums of translation and geodesic triangles in the universal cover of $\SLR$ geometry.
We prove that the angle sum $\sum_{i=1}^3(\alpha_i) \ge \pi$ for translation triangles and for geodesic triangles the angle sum can
be larger, equal or less
than $\pi$.