#### Date of Award

Winter 1999

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

David Feldman

#### Abstract

If G is a finitely generated group, then the set of all characters from G into a linear algebraic group is a useful (but not complete) invariant of G . In this thesis, we present some new methods for computing with the variety of SL2C -characters of a finitely presented group. We review the theory of Fricke characters, and introduce a notion of presentation simplicity which uses these results. With this definition, we give a set of GAP routines which facilitate the simplification of group presentations. We provide an explicit canonical basis for an invariant ring associated with a symmetrically presented group's character variety. Then, turning to the divisibility properties of trace polynomials, we examine a sequence of polynomials rn(a) governing the weak divisibility of a family of shifted linear recurrence sequences. We prove a discriminant/determinant identity about certain factors of rn( a) in an intriguing manner. Finally, we indicate how ordinary generating functions may be used to discover linear factors of sequences of discriminants.

Other novelties include an unusual binomial identity, which we use to prove a well-known formula for traces; the use of a generating function to find the inverse of a map xn ∣→ fn(x); and a brief exploration of the relationship between finding the determinants of a parametrized family of matrices and the Smith Normal Forms of the sequence.

#### Recommended Citation

Hall, Jeffrey Allan, "Using character varieties: Presentations, invariants, divisibility and determinants" (1999). *Doctoral Dissertations*. 2105.

https://scholars.unh.edu/dissertation/2105