ANGLE GEOMETRY IN THE UNIVERSAL TEICHMÜLLER SPACE.

In this paper we investigate angle geometry in the universal Teichmüller space. We construct three examples of triangles bounded by geodesic segments such that the first example has the sum of the three inner angles less than n, the second example has the sum of the three angles equal to n, and the third example has the sum of the three angles greater than π. Our result gives a negative answer to a problem raised by Zhong Li and Yi Qi. Moreover, our results indicate that the universal Teichmüuller space presents all hyperbolic, Euclidean, and spherical phenomena in angle geometry. [ABSTRACT FROM AUTHOR]