The group SL2(R) is at the cross-road of several lines of mathematicaldevelopment: for number theorists its arithmetical lattices areobjects of fundamental interest; in dynamics, it provides the simplestexample of mixing Anosov flow; for harmonic analysts SL2(R) is thestarting point of the harmonic analysis of reductive Lie groups. Inthis course, we will review the geometrical and analytic properties ofSL2(R), introduce its representation theory and its applications tohyperbolic surfaces, thereby providing a fundamental example of thetheory of resonances for hyperbolic flows developed in the parallelcourses.