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    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
    (Massey University, 2024-02-10) Salehi, Keyvan
    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.
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    Modelling of mereotopological relationships in multidimensional space : a thesis submitted for the degree of Doctor of Philosophy, School of Natural and Computational Sciences, Massey University
    (Massey University, 2021) Izadi, Azadeh
    Inferences based on spatial knowledge play an important role in human lives. Humans are easily able to deal with spatial knowledge without any need to refer to numerical computation. The field of Qualitative Spatial Representation and Reasoning (QSRR) aims to model human common sense of space. Among the various types of qualitative relationship between spatial objects, connectivity (or topology) and parthood (or mereology) serve as the most basic underlying aspects. Most current mereotopological theories are restricted to objects with the same dimension. However, sometimes spatial entities of different dimensions must be considered for many practical applications (e.g. map reading, spatial analysis). The inability of current theories to interact with entities of different dimensions has motivated the foundation of multidimensional spatial theories. However, these theories are less efficient in terms of reasoning power. Moreover, their set of introduced mereotopological relations has not been cognitively validated. This research presents a multidimensional mereotopological theory using part of and boundary part as primitive concepts. We introduce a set of nine spatial relations with the jointly exhaustive and pairwise disjoint property based on these primitives. This property allows us to develop an efficient reasoning strategy (i.e. constraint-based reasoning) which makes our approach more practical than previous works. We used automated theorem provers and finite model finders to aid the formal verification of the theory, proving its properties and generating the composition table for reasoning purposes. This work is the first multidimensional mereotopological theory that not only has properties that are verified by traditional logical deduction techniques (like the other multidimensional mereotopological theories), but that also it supports an efficient reasoning strategy that was not being available before. Furthermore, we verified the cognitive adequacy of our proposed set of relations using human subjects experiments, applying clustering and thematic analyses to empirical data. Our study is the first to pro- vide evidence for the cognitive plausibility of a multidimensional mereotopological theory (going beyond previous studies that have only shown cognitive adequacy for equidimensional mereotopological theories) supporting its closeness to human cognition. In addition, we demonstrate our multidimensional theory by applying it to a real-world scenario (i.e. a flood event).