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.
***
The draft of the thesis can be found here: honaryar_thesis_v2
no comment as of now