In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

Im Jahr 2006, in einem Papier in Compositio Titel "Bounds auf kanonische Green-Funktionen" J. Jorgenson und J. Kramer, haben optimale Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem kompakten hyperbolischen Riemannschen Fläche definiert abgeleitet. Diese Schätzungen wurden im Hinblick auf abgeleitete Invarianten aus hyperbolischen Geometrie der Riemannschen Fläche. Als Anwendung abgeleitet sie Schranken für die kanonische Green-Funktionen durch Abdeckungen und für Familien von Modulkurven. In dieser Arbeit erweitern wir ihre Methoden nichtkompakten hyperbolischen Riemann Oberflächen und leiten ähnliche Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem nichtkompakten hyperbolischen Riemannschen Fläche definiert. / In 2006, in a paper in Compositio titled "Bounds on canonical Green''s functions", J. Jorgenson and J. Kramer have derived optimal bounds for the hyperbolic and canonical Green''s functions defined on a compact hyperbolic Riemann surface. These estimates were derived in terms of invariants coming from hyperbolic geometry of the Riemann surface. As an application, they deduced bounds for the canonical Green''s functions through covers and for families of modular curves. In this thesis, we extend their methods to noncompact hyperbolic Riemann surfaces and derive similar bounds for the hyperbolic and canonical Green''s functions defined on a noncompact hyperbolic Riemann surface.