Connection blocking in lattice quotients of connected lie groups
Finite blocking is an interesting concept originating as a problem in billiard dynamics and later in the context of Riemannian manifolds. Let $(M,g)$ be a complete connected, infinitely differentiable Riemannian manifold. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a \textit{finite} set $B \subset M\setminus \{m_1, m_2 \}$ such that every geodesic segment joining $m_1$ and $m_2$ intersects $B$. $B$ is called a \textit{blocking set} for the pair $m_1,m_2 \in M$. The manifold $M$ is \textit{secure} if every pair of points in $M$ can be blocked. $M$ is \textit{uniformly secure} if the cardinality of blocking sets for all pairs of points in $M$ has a (finite) upper bound. The main blocking conjecture states that a closed Riemannian manifold is secure if and only if it is flat.Gutkin \cite{Connection blocking} initiated a similar study of blocking properties of quotients $G/\Gamma$ of a connected Lie group $G$ by a lattice $\Gamma \subset G$. Here the connection curves are the orbits of one parameter subgroups of $G$. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connection curve joining $m_1$ and $m_2$ intersects $B$. The lattice quotient $M=G/\Gamma$ is \textit{connection blockable} if every pair of points in $M$ can be blocked, otherwise we call it \textit{nonblockable}. The corresponding main blocking conjecture states that $M=G/\Gamma$ is blockable if and only if its universal cover $\tilde{G}=\mathds{R}^n$, i.e. $M$ is a torus. In this dissertation we investigate blocking properties for two classes of lattice quotients, which are lattice quotients of semisimple and solvable Lie groups.According to the Levi decomposition, every connected Lie group $G$ is a semidirect product of a solvable Lie group $R$, and a semisimple Lie group $S$. A Lie group $G=R \rtimes S$ satisfies \textit{Raghunathan's condition} if the kernel of the action of $S$ on $R$ has no compact factors in its identity component. For a such Lie group $G$, if quotients of $R$ are nonblockable then quotients of $G$ are also nonblockable. The special linear group $\textrm{SL}(n,\mathds{R})$ is a simple Lie group for $n>1$.Let $M_n= \textrm{SL}(n,\mathds{R})/\Gamma$, where $\Gamma=\textrm{SL}(n,\mathds{Z})$ is the integer lattice. We focus on $M_2$ and show that the set of blockable pairs is a dense subset of $M_2 \times M_2$, and we use this to conclude manifolds $M_n$ are nonblockable. Next, we review a quaternionic structure of $\textrm{SL}(2,\mathds{R})$ and a way for making cocompact lattices in this context. We show that the obtained lattice quotients are not finitely blockable. In the context of solvable Lie groups, we study lattice quotients of \textit{Sol}. \textit{Sol} is a unimodular solvable Lie group, with the left invariant metric $ds^2=e^{2z}dx^2+e^{2z}dy^2+dz^2$, and is one of the eight homogeneous Thurston 3geometries. We prove that all quotients of $Sol$ are nonblockable. In particular, we show that for any lattice $\Gamma \subset Sol$, the set of nonblockable pairs is a dense subset of $Sol/\Gamma \times Sol/\Gamma$.
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Electronic Theses & Dissertations
 Copyright Status
 In Copyright
 Material Type

Theses
 Authors

Bidar, Mohammadreza
 Thesis Advisors

Schmidt, Benjamin I.
 Committee Members

Parker, Thomas H.
Kulkarni, Rajesh S.
Wang, Xiaodong
 Date
 2018
 Program of Study

Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 vii, 69 pages
 ISBN

9780438277335
0438277333
 Permalink
 https://doi.org/doi:10.25335/M5KD1QQ4W