The thesis is organized as follows: First we state basic ergodic theorems in Section 2 and introduce the notation of Cesàro averages for multiple operators in Section 3. We state a general theorem in Section 3 for groups that can be represented by a finite alphabet and a transition matrix.
In the second part we show that finitely generated Fuchsian groups, with certain restrictions to the fundamental domain, admit such a representation. To develop the representation we give an introduction into Möbius transformations (Section 4), hyperbolic geometry (Section 5), the concept of Fuchsian groups and their action in the hyperbolic plane (Section 6) and fundamental domains (Section 7). As hyperbolic geometry calls for visualization we included images at various points to make the definitions and statements more approachable.
With those tools at hand we can develop a geometrical coding for Fuchsian groups with respect to their fundamental domain in Section 8. Together with the coding we state in Section 9 the main theorem for Fuchsian groups. The last chapter (Section 10) is devoted to the application of the main theorem to three explicit examples. We apply the developed method to the free group F3, to a fundamental group of a compact manifold with genus two and we show why the main theorem does not hold for the modular group PSL(2, Z).:1 Introduction
2 Ergodic Theorems
2.1 Mean Ergodic Theorems
2.2 Pointwise Ergodic Theorems
2.3 The Limit in Ergodic Theorems
3 Cesàro Averages of Sphere Averages
3.1 Basic Notation
3.2 Cesàro Averages as Powers of an Operator
3.3 Convergence of Cesàro Averages
3.4 Invariance of the Limit
3.5 The Limit of Cesàro Averages
3.6 Ergodic Theorems for Strictly Markovian Groups
4 Möbius Transformations
4.1 Introduction and Properties
4.2 Classes of Möbius Transformations
5 Hyperbolic Geometry
5.1 Hyperbolic Metric
5.2 Upper Half Plane and Poincaré Disc
5.3 Topology
5.4 Geodesics
5.5 Geometry of Möbius Transformations
6 Fuchsian Groups and Hyperbolic Space
6.1 Discrete Groups
6.2 The Group PSL(2, R)
6.3 Fuchsian Group Actions on H
6.4 Fuchsian Group Actions on D
7 Geometry of Fuchsian Groups
7.1 Fundamental Domains
7.2 Dirichlet Domains
7.3 Locally Finite Fundamental Domains
7.3.1 Sides of Locally Finite Fundamental Domains
7.3.2 Side Pairings for Locally Finite Fundamental Domains
7.3.3 Finite Sided Fundamental Domains
7.4 Tessellations of Hyperbolic Space
7.5 Example Fundamental Domains
8 Coding for Fuchsian Groups
8.1 Geometric Alphabet
8.1.1 Alphabet Map
8.2 Transition Matrix
8.2.1 Irreducibility of the Transition Matrix
8.2.2 Strict Irreducibility of the Transition Matrix
9 Ergodic Theorem for Fuchsian Groups
10 Example Constructions
10.1 The Free Group with Three Generators
10.1.1 Transition Matrix
10.2 Example of a Surface Group
10.2.1 Irreducibility of the Transition Matrix
10.2.2 Strict Irreducibility of the Transition Matrix
10.3 Example of PSL(2, Z)
10.3.1 Irreducibility of the Transition Matrix
10.3.2 Strict Irreducibility of the Transition Matrix
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:76431 |
| Date | 04 November 2021 |
| Creators | Drygajlo, Lars |
| Contributors | Pogorzelski, Felix, Rademacher, Hans-Bert, Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:masterThesis, info:eu-repo/semantics/masterThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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