Dissertations, Theses, and Capstone Projects

Date of Degree

9-2020

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Ara Basmajian

Committee Members

Melvyn Nathanson

Dragomir Saric

Robert Suzzi Valli

Subject Categories

Geometry and Topology | Mathematics | Physical Sciences and Mathematics

Keywords

hyperbolic geometry, reciprocal geodesics, geometric group theory, analytic combinatorics, recurrence relations

Abstract

In this thesis we obtain the growth rates for conjugacy classes of reciprocal words for triangle groups of the form G = Z2 ∗ H where H is finitely generated and does not contain an order 2 element. We explore cases where H is infinite cyclic and finite cyclic. The quotient O = H/G is an orbifold and contains a cone point of order 2, due to the first factor Z2 in the free product G. The reciprocal words in G correspond to geodesics on O which pass through the order 2 cone point on O. We use methods from analytic combinatorics to count these words and to analyze the asymptotic behavior of their conjugacy classes with respect to the word length in the group for chosen generators of minimal length in G. We specifically use recurrence relations, and techniques for obtaining the closed forms of these relations, in order to study the growth of these geometric objects. With these methods we are able to describe the asymptotic behavior of the conjugacy classes of the primitive reciprocal geodesics in the groups as well. We find exponential asymptotics in all of the cases studied here and can compare the different bases between them.

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