Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.
Deep Dive into Hyperbolic Geometry and Distance Functions on Discrete Groups.
Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky–Mozes–Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.
This thesis reflects my interest in the history of mathematics, and in areas of mathematics which combine geometry and algebra. The historical and mathematical sections of this work can be read independently. One of the main themes of the history, though, is how in the nineteenth century a new understanding of mathematical space developed, in which groups became vital to geometry. The idea of using algebraic concepts to investigate geometric structures is present throughout the work.
Chapter 1 is a short history of non-Euclidean geometry. This chapter is a synthesis of my readings of mainly secondary sources. Historians of mathematics have traditionally adopted an ‘internalist’ approach, explaining changes to the content of mathematics only in terms of factors such as the imperative to generalise, or the desire to remove inconsistencies. An internalist narrative is certainly necessary for a satisfactory history of mathematics, but it is not sufficient. First, the history of mathematics is part of broader intellectual history. Euclidean geometry held great philosophical prestige, and non-Euclidean geometry challenged fundamental assumptions about the nature of space and the truth value of mathematics. Second, there are social factors in the history of mathematics. Mathematicians themselves are part of a community, the mathematical community. The structure of this community in the nineteenth century helps to explain the reception and dissemination of non-Euclidean geometry. The sources consulted for this chapter each adopted one or more of the approaches to the history of mathematics outlined here, but none of them seemed to discuss all the relevant issues.
Chapter 2 begins the strictly mathematical part of this thesis. One aim of this chapter is to provide a deeper understanding of some of the mathematics discussed in Chapter 1. To this end, each of the main models of hyperbolic geometry is presented, and important results for each model proved. The second aim of Chapter 2 is to describe thoroughly the way in which the action of the group P SL(2, Z) induces a tesselation of the upper half-plane. Most of the material in Chapter 2 is selected and adapted from Chapters 3-6 of Ratcliffe [21].
Chapter 3 poses a question about the upper half-plane, and then provides the theory needed to frame the question precisely and answer it in a variety of settings. Suppose z is a point in the upper half-plane, and γz the result of the action of an element γ ∈ P SL(2, Z) on the point z. The geodesic segment joining z and γz has a finite length, and crosses a finite number of tiles in the tesselation of the upper half-plane. The question is whether or not there is any relationship between the length of the geodesic segment, and the number of tiles it crosses. We begin by defining a symmetric space. Then, if X is a symmetric space, and Γ a discrete group of isometries of X, we define two distance functions on the group Γ. The first is the geometric distance function, and is induced by the distance function on the space Chapter 1
A History of Non-Euclidean Geometry I entreat you, leave the doctrine of parallel lines alone; you should fear it like a sensual passion; it will deprive you of health, leisure and peace-it will destroy all joy in your life.
from a letter by Farkas Bolyai to his son János. 1
This chapter is a short history of non-Euclidean geometry. The intellectual history of developments internal to mathematics is presented, together with discussion of relevant social and philosophical trends within the mathematical community and in wider contexts. Chapter 2 will provide a rigorous and modern treatment of many of the mathematical terms and results referred to here. In order to properly acknowledge sources, the referencing style adopted for Chapter 1 is different from that in the rest of this thesis. The discovery and acceptance of non-Euclidean geometry forms a crucial strand of the history of mathematics. For thousands of years, Euclidean geometry held great prestige. However, there were small doubts about the parallel axiom. After numerous attempts to validate this axiom, in the nineteenth century there was a shift to advancing alternative geometries. The coming-together of developments in many areas of mathematics aided the acceptance of non-Euclidean geometry by the mathematical community. In this process mathematicians were forced to reconsider and alter their ideas about the nature of mathematics and its relationship with the real world.
Although Egyptian and Babylonian mathematicians had posed and solved geometric problems, geometry only acquired logical structure with the Greeks. 3 Greek 1 Quoted in B. A. Rosenfeld. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Springer-Verlag, 1988, p. 108. 2 Quoted in Jeremy Gray. Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Clarendon Press, 1989, p. 107. 3 Carl B. Boyer. A History of Mathematics. John Wiley & Sons,
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