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Ergodicity of the geodesic flow on quotient surfaces of the hyperbolic plane

Isabel Gomes Gerber

2020
unpublished

The goal of my thesis is to show the ergodicity of the geodesic flow on quotient spaces of the hyperbolic plane Γ \ H, where Γ is a lattice. This statement is presented and proven in the last section of chapter 3. To be able to understand all the concepts needed, we start by introducing the hyperbolic plane H in chapter 1 and point out how its geometry differs from Euclidean geometry. In particular, we demonstrate how hyperbolic distance is defined and show its consequences. For instance, we
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... l see that geodesics in the hyperbolic plane consist of vertical lines and semicircles with centre on R. We will also be interested in studying Möbius transformations, which do not alter hyperbolic distances, angles or hyperbolic areas. Furthermore some fundamental properties of hyperbolic geometry will be shown, such as the Gauss-Bonnet Theorem. Chapter 2 starts by showing various characteristics of the projective special linear group, PSL2(R), such as the identification between PSL2(R) and T^1H, the unit tangent bundle of H, or the fact that PSL2(R) is a closed linear group. The reason why this is useful is that since the geodesic flow on the hyperbolic plane is a function on T^1H we can also define the geodesic flow as a function on PSL2(R), which will be done in chapter 3. We will also derive a metric on PSL2(R). This will be done in a more general way by defining a metric on closed linear groups G. Afterwards we consider properties of Fuchsian groups and introduce the notion of fundamental regions. This will be important since we want Γ to be a Fuchsian group whose fundamental domains have finite measure. As mentioned before we start chapter 3 by defining the geodesic flow on T^1H as well as on PSL2(R). The same can be done for the horocycle flow. In order to define the geodesic flow on the quotient space Γ \ PSL2(R) we first demonstrate the identifications T^1(Γ \ H) = Γ \ (T^1H) = Γ \ PSL2(R) and use the definition of the geodesic flow on T^1(Γ \ H). Our last step before examining the ergodicity of the geodesic flow [...]

doi:10.25365/thesis.60750
fatcat:2fggtkukffapxibxbblsqwozdu