Graduate Program

Mathematics

Degree Name

Master of Arts (MA)

Semester of Degree Completion

Summer 2021

Thesis Director

Grant Lakeland

Thesis Committee Member

Gregory Galperin

Thesis Committee Member

Bogdan V. Petrenko

Abstract

This thesis concerns the study of the Cheeger constant of two related hyperbolic Riemann surfaces. The first surface R is formed by taking the quotient U2/Γ(4), where U2 is the upper half-plane model of the hyperbolic plane and Γ(4) is a congruence subgroup of PSL2(Z), an isometry group of U2 . This quotient is shown to form a Riemann surface which is constructed by gluing sides of a fundamental domain for Γ(4) together according to certain specified side pairings. To form the related Riemann surface R' , we follow a similar procedure, this time taking the quotient U2/G, where G is an index 2 subgroup of Γ(4). For both R and R' , we provide an estimate of the Cheeger constant using a procedure given in [2]. The Cheeger constant is believed to be the same for both surfaces.

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