The Dimension of the Restricted Hitchin Component for Triangle Groups

Author

Elise Anne Weir, University of TennesseeFollow

Orcid ID

http://orcid.org/0000-0003-4681-8839

Date of Award

8-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Morwen B. Thistlethwaite

Committee Members

Michael W. Berry, Luis R. A. Finotti, Marie Jameson

Abstract

Given positive integers p, q, r satisfying 1/p + 1/q + 1/r < 1, the hyperbolic triangle group T(p,q,r) is the group of orientation-preserving isometries of a tiling of the hyperbolic plane by triangles congruent to a geodesic triangle with angles π/p, π/q, and π/r. We will examine representations of triangle groups in the Hitchin component, a topologically connected component of the representation variety where representations are always discrete and faithful.We begin by giving a formula for the dimension of a subset of the Hitchin component of an arbitrary hyperbolic triangle T(p, q, r) for general degree n > 2. Depending on whether n is even or odd, we will consider only those Hitchin representations whose images lie in Sp(2m) or SO(m,m + 1), respectively. We call the space of representations satisfying this criterion the restricted Hitchin component.We then provide two new families of representations of the specific triangle group T(3,3,4) into SL(5,R); the image groups of these families are each shown to be Zariski dense in SL(5,R). Further, we consider a restriction to a surface subgroup of finite index in T(3,3,4). For each family, we will demonstrate the existence of a subsequence of representations whose images are pairwise non-conjugate in SL(5,Z) when restricted to a surface subgroup.