Monday, August 21, 2023

11:00 a.m.

Zoom Web Conference

PhD Candidate: Pouya Honaryar

Supervisor: Kasra Rafi

Thesis title: Lattice point counts in Teichmüller space and negative curvature

In this thesis we present two different, but related, results; one in the setting of

Teichmüller theory, and the other in the setting of negative curvature. For the first

result, let be a pseudo-Anosov homeomorphism of a compact orientable surface

Sg, and let L denote the axis of the action of on the Teichmüller space of Sg,

denoted by Tg. In Chapter 3 we obtain asymptotics for the number of translates

of L that intersect a Teichmüller ball of radius R centered at a fixed X 2 Tg, as

R ! 1. 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 fM of M, fix an

element id 6= in the fundamental group of M, and denote the set of elements

in that are conjugate to by Conj . In Chapter 4 we obtain asymptotics for the

number of Conj –orbits of y that lie in a ball of radius R centered at x, as R ! 1.

If M is two-dimensional, or of dimension n 3 and curvature bounded above by

1 and below by (n1

n2 )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 underlying

ideas shared by both results.

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The draft of the thesis can be found here: honaryar_thesis_v2

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