Hyperbolic hypergeometric monodromy groups and geometric finiteness

Abstract

The subject of this thesis is hyperbolic hypergeometric monodromy groups.

I studied the monodromy groups $H(\alpha,\beta)$ for $_nF_{n-1}$ hypergeometric differential equation $Du=0$, where

$D=(\theta+\beta_1-1)\cdots(\theta+\beta_n-1)-z(\theta+\alpha_1)\cdots(\theta+\alpha_n)$ and $\theta = z \frac{d}{dz}, \alpha,\beta \in \mathbb{Q}^n$.

$H(\alpha,\beta)$ is generated by the local monodromies $A,B,C$ which correspond to $0,1,\infty$ respectively. Let $G$ is the zariski closure of $H(\alpha,\beta)$. In the self-dual case, $G$ is either finite, $Sp(n)$, or $O(n)$. I focus on $G$ orthogonal and of signature $(n-1,1)$ over $\mathbb{R}$.

The first result is that the odd hyperbolic hypergeometric monodromy groups $H(\alpha,\beta)$ are of infinite index in $G(\mathbb{Z})$. In this case $H(\alpha,\beta)$ is called 'thin', which is equivalent to saying that the fundamental domain has infinite volume. It was shown that there are three families of odd hyperbolic hypergeometric monodromy groups, and we show that $H(\alpha,\beta)$ is thin for all but one case by showing that a finite index subgroup of $H(\alpha,\beta)$ is commensurable with a reflection subgroup of an infinite index subgroup of $G(\mathbb{Z})$. In the remaining case, I used an algorithm from \cite{FMS} to show that the fundamental domain for a finite index reflection subgroup of $H(\alpha,\beta)$ is thin. This covers the second chapter.

In the third chapter, I investigated the full reflection subgroup of $G(\mathbb{Z})$, the $2$-reflection subgroup containing all $2$-reflections, and $H(\alpha,\beta)$. When $\Gamma$ be one of them, I looked at limit set $\Lambda(\Gamma)$, the exponent of divergence $\delta(\Gamma)$ of the Poincare series, and the bottom of the spectrum of the Laplacian on $L^2(\Gamma\backslash\mathbb{H}^{n-1})$. The second result is 'geometric finiteness' for discrete groups generated by finitely many reflections, which means its fundamental domain has finitely many faces. It follows that $H(\alpha,\beta)$'s which were shown to be thin in \cite{FMS} and in the second chapter are not only thin, but have 'sparse' limit sets.

In the last chapter, the first example for which the monodromy group is geometrically finite and thin is given. I found a fundamental domain using numerical computation, and showed that it is the fundamental domain for $H(\alpha,\beta)$ using Poincare's polyhedron theorem.