Lattice Point Counts in Teichmuller Space and Negative Curvature

In this thesis we present two different, but related, results; one in the setting of Teichmuller theory, and the other in the setting of negative curvature. For the first result, let $\gamma$ be a pseudo-Anosov homeomorphism of a compact orientable surface $S_g$, and let $L_\gamma$ denote the axis of the action of $\gamma$ on the Teichmuller space of $S_g$, denoted by $\mathcal{T}_g$. In Chapter 3 we obtain asymptotics for the number of translates of $L_\gamma$ that intersect a Teichmuller ball of radius $R$ centered at a fixed $X \in \mathcal{T}_g$, as $R \rightarrow \infty$. For the second result, let $M$ be a compact closed manifold of variable negative curvature. We fix two points $x, y$ in the universal cover $\widetilde{M}$ of $M$, fix an element $\mathrm{id} \neq \gamma$ in the fundamental group $\Gamma$ of $M$, and denote the set of elements in $\Gamma$ that are conjugate to $\gamma$ by $\mathrm{Conj}_\gamma$. In Chapter 4 we obtain asymptotics for the number of $\mathrm{Conj}_\gamma$--orbits of $y$ that lie in a ball of radius $R$ centered at $x$, as $R \rightarrow \infty$. If $M$ is two-dimensional, or of dimension $n \geq 3$ and curvature bounded above by $-1$ and below by $-(\frac{n-1}{n-2})^2$, we find a power saving error term for this count. Since the two results are written in different settings, their similarities might be hidden at first glance. This is why we included Chapter 2, in which we present a unified approach to both results in the setting of constant negative curvature. Writing the arguments in this simple setting helps us emphasize the similarities between the two results.

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