#### 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.*

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*#### Recommended Citation

Weir, Elise Anne, "The Dimension of the Restricted Hitchin Component for Triangle Groups. " PhD diss., University of Tennessee, 2018.

https://trace.tennessee.edu/utk_graddiss/5089

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