  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.
