## The (5, p)-arithmetic hyperbolic lattices in three dimensions : a dissertation in Mathematics, presented to the Massey University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Loading...

##### Date

2024-02-10

##### DOI

##### Open Access Location

##### Authors

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

Massey University

##### Rights

The Author

##### Abstract

The group $Isom^+(\mathbb{H}^3)\cong PSL(2,\mathbb{C})$ contains an unlimited number of lattices of orientation-preserving isometries of hyperbolic 3-space (equivalently Kleinian groups of finite co-volume) that may be produced by using two elements of finite orders $p$ and $q$ as generators. For example, all but a finite number of $(p, 0)$-$(q, 0)$ orbifold Dehn surgery on any of the infinite number of hyperbolic two-bridge links (or knots if $p = q$) would have (orbifold) fundamental groups that are such uniform (co-compact) lattices. However, it was demonstrated in \cite{MM} that, up to conjugacy, two elements of finite order could generate only a finite number of arithmetic lattices. In fact, it is proved in \cite{MM} that there are only a finite number of {\em nearly arithmetic} groups, that is groups generated by two elements of finite order that are discrete subgroups of arithmetic groups and are not free on the two generators.
The main result of this thesis is the determination of all the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $5$ and an element of order $p\geq 2$, along with the determination of all the associated nearly arithmetic groups. These groups $\Gamma$ will have a presentation of the form
\[ \Gamma\cong\langle f,g: f^5=g^p=w(f,g)=\cdots=1 \rangle \]
In this Thesis, we find that necessarily
\begin{itemize}
\item $p\in \{2,3,4,5\}$
\item The total degree of the invariant trace field
\[ k\Gamma=\mathbb{Q}(\{tr^2(h):h\in\Gamma\})\]
is at most $6$ and at most $4$ for lattices.
\item Each orbifold is either a two bridge link of slope $r/s$ surgered with $(5,0)$, $(p,0)$ orbifold Dehn surgery or a Heckoid group with rational slope $r/s\in [0,1]$ and $w(f,g)=(w_{r/s})^r$ with $r\in \{2,3,4,5\}$, and $w_{r/s}$ is a Farey word - described later.
\end{itemize}
For each such group, we find a discrete and faithful representation in $PSL(2,\mathbb{C})$, identify the rational slope $r/s$ and identify the associated number theoretic data.

##### Description

##### Keywords

Arithmetical algebraic geometry, Hyperbolic spaces, Lattice theory, Set theory