Finite Quotients of Triangle Groups

Frankie Chan, Purdue University

Abstract

Extending an explicit result from Bridson–Conder–Reid [1], this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups ∆ and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(2, Fq) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of ∆ and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between ∆ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.

Degree

Ph.D.

Advisors

McReynolds, Purdue University.

Subject Area

Mathematics

COinS