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\textbf{18.786 Number Theory II\hfill Spring~2024}\\\vspace{2pt}
\textbf{Problem Set \#2\hfill Due: 2/25/2024}\vspace{-8pt}
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\subsection*{Description}

These problems are related to Lectures 3-5.
Your solutions should be written up in latex and submitted as a pdf-file on \href{https://www.gradescope.com/courses/730173}{Gradescope} before midnight on the date due.

Collaboration is permitted/encouraged, but you must identify your collaborators or the name of your group on \href{https://psetpartners.mit.edu}{pset partners}, and any references you consulted. If there are none write ``\textbf{Sources consulted: none}'' at the top of your solution.
\medskip

\noindent
\textbf{Instructions:} In the problems below we reference terms and definitions from earlier problems, so be sure to read through the problems in order.  You may use results proved in an earlier problem in any later problem (even if you do not choose to solve the earlier problem).
Pick any combination of problems to solve that sum to 100 and write up your answers in latex (be sure to include Problem 4, which is a survey).

\medskip
\noindent
Notation: $\H:=\{z\in \C: \Im(z)>0\}$, $\D:=\{z\in \C:|z|<1\}$, $Z(\Gamma):=\Gamma\cap\{\pm 1\}$.

\subsection*{Problem 1. Classification of Fuchsian groups (50 points)}
As in Problem 2 of Problem Set 1, we call $z\in \P^1(\C) := \C\cup \{\infty\}$ a \defi{limit point} of a group $\Gamma\subseteq \SL_2(\R)$ if there is a $w\in \P^1(\C)$ and an infinite sequence of distinct $\gamma_n\in \Gamma$ such that $\lim \gamma_n w = z$, and we use $\Lambda(\Gamma)$ to denote the set of of all limit points of $\Gamma$.

Recall that a \defi{Fuchsian group} is a subgroup $\Gamma\le \SL_2(\R)$ that satisfies any of the equivalent properties listed in part (a) of Problem 2 on Problem Set 1; in particular, $\Gamma$ is discrete and its limit set $\Lambda(\Gamma)$ is a closed subset of $\P^1(\R):=\R\cup \{\infty\}$.
\defi{Elliptic points} are elements of $\H$ fixed by an elliptic $\gamma\in \Gamma$, \defi{cusps} are elements of $\P^1(\R)$ fixed by a parabolic $\gamma\in \Gamma$, and \defi{hyperbolic points} are elements of $\P^1(\R)$ fixed by a hyperbolic $\gamma\in \Gamma$.

Recall that in a topological space $X$ a point $x\in X$ is a \defi{limit point} of a set $S\subseteq X$ if every punctured neighborhood of $x$ contains a point in $S$.
A set $S\subseteq X$ is \defi{perfect} if it is equal to its set of limit points and \defi{nowhere dense} if its interior is empty, equivalently, every punctured neighborhood of $x\in S$ contains a point not in $S$.  Let $\Gamma$ be a Fuchsian group.

\begin{enumerate}[{\bf(a)}]
\item Show that every $x\in \P^1(\R)$ fixed by an element of $\Gamma-Z(\Gamma)$ lies in $\Lambda(\Gamma)$; in other words, $\Lambda(\Gamma)$ contains all cusps and hyperbolic points of $\Gamma$. 
\item Show that $\Gamma$ is countable and $\H$ contains uncountably many non-elliptic points.
\item Show that for any non-elliptic point $w\in \H$ the set $\Lambda(\Gamma)$ is equal to the set of limit points of the $\Gamma$-orbit of $w$, and in particular is $\Gamma$-invariant.
\item Show that if $x\in \Lambda(\Gamma)$ is fixed by a hyperbolic $\gamma\in \Gamma$ and $y\in \Lambda(\Gamma)$ is not fixed by $\gamma$ then either $\lim_{n\to\infty}\gamma^n y = x$ or $\lim_{n\to\infty}\gamma^{-n}y = x$, so $x$ is a limit point of $\Lambda(\Gamma)$.
\item Show that if $x\in \Lambda(\Gamma)$ is not a hyperbolic point and $\Gamma$ contains a hyperbolic $\gamma$, then $x$ is a limit of hyperbolic points (hint: apply \textbf{(c)} to a point fixed by $\gamma$)
\item Show that if $\#\Lambda(\Gamma)>2$ then $\Gamma$ contains a hyperbolic element $\gamma$ and $\Lambda(\Gamma)$ is perfect, and therefore uncountable.
\item Show that if $\Lambda(\Gamma)\ne \P^1(\R)$ is perfect then $\Lambda(\Gamma)$ is nowhere dense.  Use this to show that $\Lambda(\SL_2(\Z))=\P^1(\R)$.
\end{enumerate}

\noindent
It follows from \textbf{(f)} and \textbf{(g)} that there are three kinds of Fuchsian groups:
\begin{itemize}
\setlength{\itemsep}{-2pt}
\item \defi{elementary}: $\#\Lambda(\Gamma) \le 2$.
\item \defi{the first kind}: $\Lambda(\Gamma)=\P^1(\R)$.
\item \defi{the second kind}: $\Lambda(\Gamma)$ is perfect and nowhere dense.
\end{itemize}

\subsection*{Problem 2. Finitely generated Fuchsian groups (50 points)}

Let $\Gamma$ be a Fuchsian group, $w\in \H$ a non-elliptic point, and $F_w$ the corresponding Dirichlet domain.
Recall that the boundary of $F_w$ is a union of geodesic segments of the form $L_\gamma:= F_w\cap \gamma F_w$ called \defi{sides}; we require geodesic segments to have nonzero (possibly infinite) length (so if $L_\gamma$ is a point it is not a side).

If $F_w$ has only finitely many sides then $\Gamma$ is called \defi{geometrically finite} (this does not depend on the choice of $w$).  Problem 3 of Problem Set 1 proves that if $\Gamma$ is geometrically finite then it is finitely generated.  The goal of this problem is to prove the converse.

As in Problem 3 of Problem Set 1, we split sides of the form $L_\gamma$ with $\gamma^2=\pm 1$ into two sides that intersect at the midpoint of $L_\gamma$ and group sides into pairs
that are $\Gamma$-translates (and not $\Gamma$-equivalent to any other sides).
Let $G(\Gamma)$ be the subset of $\Gamma$ consisting of elements $\gamma$ that appear in a pair $(L,\gamma L)$.
As proved in Problem Set 1, the set $G(\Gamma)$ generates $\Gamma$.
\bigbreak

\noindent
Now suppose that $\Gamma$ is finitely generated.  Out goal is to show $F_w$ has finitely many sides.

\begin{enumerate}[{\bf(a)}]
\setlength{\itemsep}{2pt}
\item Show that if $\Gamma$ is finitely generated then any generating set for $\Gamma$ contains a finite subset that generates $\Gamma$.
Let $\gamma_1,\ldots\gamma_n$ be a minimal subset of $G(\Gamma)$ generating $\Gamma$ and let $L_1,\ldots,L_n$ be corresponding sides so that $(L_i,\gamma_i L_i)$ is a side pair for $1\le i\le n$.

\item Show that there exists a hyperbolic ball $B_w$ about $w$ whose interior contains arcs of positive length on all of the sides $L_1,\ldots,L_n,\gamma_1 L_1,\ldots \gamma_n L_n$, whose boundary contains no vertices of $F_w$ and is not tangent to any side of $F_w$.
Then show that only finitely many sides of $F_w$ intersect $B_w$ and that $\Gamma B_w$ is connected.

\item Let $s_1,\ldots,s_m$ be the arcs that make up $\partial B_w\cap F_w$.  Pick an arc $s_i$ and let $e$ be one of its endpoints.  Show that for some $\gamma\in G(\Gamma)$ the point $\gamma e$ is an endpoint of some $s_j$, so that $s_i\cup \gamma^{-1}s_j$ is a connected path.
Show that by continuing in this fashion in both directions one can construct a connected path $C_i$ composed of $\Gamma$-translates of the $s_j$ that separates $\H$ into two disjoint sets, and that $\omega_i C_i=C_i$ for some $\omega_i\in \Gamma-Z(\Gamma)$.  We thus obtain $C_i$ and $\omega_i$ for each $1\le i\le m$.

\item Let $E_1,\ldots, E_m$ be the components of $F_w-B_w$ adjacent to $s_1,\ldots, s_m$.  Show that every side of $F_w$ disjoint from $B_w$ intersects one of the $E_i$, so that it suffices to show that each $E_i$ intersects only finitely many sides of $F_w$.  Then show that if no element of $\Lambda(\Gamma)$ is a limit point of $E_i$ then $E_i$ can intersect only finitely many sides of $F_w$.

\item Show that each path $C_i$ of part \textbf{(c)} separates the interior of $E_i$ from $\Gamma w$, so that any limit point of $E_i$ in $\Lambda(\Gamma)$ is an endpoint of $C_i$, and show that $C_i$ has 0, 1, or 2 endpoints, depending on whether $\omega_i$ is elliptic, parabolic, or hyperbolic, respectively.

\item Show that if $\omega_i$ is hyperbolic then $E_i$ has no limit points in $\Lambda(\Gamma)$.

\item Show that if $\omega_i$ is parabolic and $\Gamma$ is not elementary then $E_i$ lies inside $C_i$ and intersects $F_w$ in only finite many sides.  Then show that if $\Gamma$ is elementary then $F_w$ has finitely many sides, and conclude that in every case $F_w$ has finitely many sides.

\end{enumerate}

\subsection*{Problem 3. Fuchsian groups of the first kind (50 points)}
For a Fuchsian group $\Gamma$ we use $X_\Gamma$ to denote the Riemann surface $\Gamma\backslash \H_\Gamma^*$, where $\H_\Gamma^*$ is the topological space obtained by adjoining the cusps of $\Gamma$ to $\H$ (and giving them open neighborhoods), and $X_\Gamma$ is equipped with the complex structure defined in \cite[\S 1.8]{miyake}.

Recall that under the measure $\mu$ induced by the hyperbolic metric every fundamental domain $F_\Gamma$ for $\Gamma$ has the same area, and we say that $\Gamma$ is \defi{cofinite} if this $\mu(F_\Gamma)$ is finite.
Siegel's theorem \cite[\S 1.9.1]{miyake} implies that $\Gamma$ is cofinite if and only if $X_\Gamma$ is compact.

In Problem 1 (and many texts), a Fuchsian group $\Gamma$ of the \defi{first kind} is one that satisfies $\Lambda(\Gamma)=\P^1(\R)$.
Miyake \cite[p.\,28]{miyake} uses this term to indicate that $X_\Gamma$ is compact.  The goal of this problem is to understand how these two notions are related.

Let $\Gamma$ be a Fuchsian group.

\begin{enumerate}[{\bf(a)}]

\item Show that if $\Gamma'\le \Gamma$ then $\mu(F_\gamma')=[\Gamma\colon \Gamma']\mu(F_\Gamma)$ and conclude that a subgroup of a cofinite Fuchsian group is cofinite if and only if it is of finite index.

\item Show that if $\Gamma$ is cofinite then it is geometrically finite, hence finitely generated.

\item Show that if $\Gamma$ is cofinite then $\Lambda(\Gamma)=\P^1(\R)$.

\item Show that elementary Fuchsian groups are geometrically finite but not cofinite.

\item Show that for geometrically finite Fuchsian groups, if $\Lambda(\Gamma)=\P^1(\R)$ then $\Gamma$ is cofinite.

\item Assume $\Gamma$ is not elementary, and let $H(\Gamma)$ be the (hyperbolic) convex hull of $\Lambda(\Gamma)$ in $\H$, and let $D(\Gamma)$ be a Dirichlet domain.  Show that $H(\Gamma)\cap D(\Gamma)$ has finite area if and only if $\Gamma$ is geometrically finite, and $H(\Gamma)=\H$ if and only if $\Lambda(\Gamma)=\P^1(\R)$.

\item Prove that for finitely generated Fuchsian groups $\Gamma$, the Riemann surface $X_\Gamma$ is compact if and only if $\Lambda(\Gamma)=\P^1(\R)$.
\end{enumerate}

\begin{remark}
In Poincar\'e's original exposition, Fuchsian groups were assumed to be finitely generated.  Some authors still make this assumption (possibly implicitly), in which case the two definitions of \defi{first kind} we have been considering are equivalent.
However, there are infinitely generated non-cofinite Fuchsian groups for which $\Lambda(\Gamma)=\P^1(\R)$; see \cite{basmajian}.
\end{remark}

\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast", 5=``just right") and the novelty of the material to you (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
2/20 & Arithmetic groups & & & & \\\hline
2/21 & Automorphic forms & & & & \\\hline 
\end{tabular}
\end{center}

\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.

\begin{thebibliography}{99}
\bibitem{basmajian} A. Basmajian, \href{http://www.ams.org/journals/tran/1993-336-01/S0002-9947-1993-1087051-2/}{\textit{Hyperbolic structures for surfaces of finite type}}, Trans. Amer. Math. Soc. \textbf{336} (1993), 421--444.
\bibitem{miyake} T. Miyake, \href{https://doi.org/10.1007/3-540-29593-3}{\textit{Modular forms}}, Springer, 2006.
\end{thebibliography}

\end{document}