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 real 2 x 2 matrices with unit determinant, as, a Thurston geometry denoted by P of SL2(R)~ geometry. We prove that the angle sum ?3i =1(?i) ? ? for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than ?.