A $(p,q,r)$-complex hyperbolic triangle group is a group generated by complex reflections across complex geodesics meeting at angles $\pi/p, \pi/q, \pi/r$. In this thesis, we study the algebraic aspects of complex hyperbolic triangle groups and answer questions related to their discreteness. In part I, we classify all complex hyperbolic triangle groups by types. The type of a complex hyperbolic triangle group is defined according to the ellipticity of the two words $I_1I_3I_2I_3$ and $I_1I_2I_3$. Algebraically, the type is decided by which trace of the words would enter the deltoid first. We use algebraic technique to show directly the type is determined by a particular polynomial in the cosines of the angles with integer coefficients. In part II, we study $(p,q,r;n)$-triangle groups, which are $(p,q,r)$-triangle groups with the additional constraint $I_1I_3I_2I_3$ has order $n$. In particular, we use algebraic techniques developed in Part I to show that for a discrete $(p,q,r;n)$-triangle group with $p>22$, $I_1I_2I_3$ must be finite-order regular elliptic.
Wang, Yuhan,
"On the Algebraic Aspects of Complex Hyperbolic Triangle Groups"
(2020).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/8q55-ck20
