memoire-m2

diff --git a/references.bib b/references.bib
@@ -23,6 +23,44 @@
   year     = {2023},
 }
 
+@article{funar,
+  author    = {Louis Funar},
+  doi       = {10.1090/S0002-9939-2010-10555-5},
+  journal   = {Proceedings of the American Mathematical Society},
+  language  = {English},
+  number    = 1,
+  pages     = {375--382},
+  title     = {Two Questions on Mapping Class Groups},
+  volume    = {139},
+  year      = {2011},
+}
+
+@article{franks-handel,
+  author    = {Franks, John and Handel, Michael},
+  doi       = {10.1090/S0002-9939-2013-11556-X},
+  issn      = {0002-9939},
+  journal   = {Proceedings of the American Mathematical Society},
+  number    = 9,
+  pages     = {2951--2962},
+  title     = {Triviality of Some Representations of $\operatorname{MCG}(S_g)$ in $GL(n, \mathbb{C})$, $\operatorname{Diff}(S^2)$ and $\operatorname{Homeo}(\mathbb{T}^2)$},
+  volume    = {141},
+  year      = {2013},
+}
+
+@article{kasahara,
+  author    = {Kasahara,  Yasushi},
+  doi       = {10.1090/tran/9037},
+  issn      = {1088-6850},
+  journal   = {Transactions of the American Mathematical Society},
+  month     = oct,
+  number    = 2,
+  pages     = {1183--1218},
+  publisher = {American Mathematical Society (AMS)},
+  title     = {Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces},
+  volume    = {377},
+  year      = {2023},
+}
+
 @incollection{julien,
   author    = {Marché, Julien},
   booktitle = {Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev},

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -61,7 +61,7 @@
   \(\Mod(S) \actson H_k(S, \mathbb{R})\)
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:symplectic-rep}
   The symplectic representation.
 \end{example}
 

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,15 +1,16 @@
 \chapter{Relations Among Twists \& Wajnryb's Presentation}
 
-\begin{proposition}[Lantern relation]
-\end{proposition}
+\begin{example}
+  Lantern relation
+\end{example}
 
-\begin{corollary}
-  If \(g \ge 3\) then the Abelianization \(\Mod(S_g)^\ab =
-  \mfrac{\Mod(S_g)}{[\Mod(S_g), \Mod(S_g)]}\) is trivial. In other words,
-  \(\Mod(S_g) = \Mod(S_g)'\) is a perfect group for \(g \ge 3\).
-\end{corollary}
+\begin{proposition}\label{thm:trivial-abelianization}
+  If \(g \ge 3\) then the Abelianization \(\Mod(S_g^b)^\ab =
+  \mfrac{\Mod(S_g^b)}{[\Mod(S_g), \Mod(S_g)]}\) is trivial. In other words,
+  \(\Mod(S_g) = [\Mod(S_g), \Mod(S_g)]\) is a perfect group for \(g \ge 3\).
+\end{proposition}
 
-% TODO: Explain to get the groups in the low-genus settings we use explicit
+% TODO: Explain how to get the groups in the low-genus settings we use explicit
 % presentations of Mod(S_g)
 \begin{center}
   \begin{tabular}{ r|c|l }
@@ -23,17 +24,17 @@
 
 % TODO: Look for a proof of this? In any case, cite a reference
 \begin{proposition}\label{thm:commutator-is-perfect}
-  Given \(b \ge 0\) the commutator subgroup \(\Mod(S_2^b)' = [\Mod(S_2^b),
-  \Mod(S_2^b)]\) is perfect -- i.e. \(\Mod(S_2^b)^{(2)} = [\Mod(S_2^b)',
-  \Mod(S_2^b)']\) is the whole of \(\Mod(S_2^b)'\).
+  The commutator subgroup \(\Mod(S_2^b)' = [\Mod(S_2^b), \Mod(S_2^b)]\) is
+  perfect -- i.e. \(\Mod(S_2^b)^{(2)} = [\Mod(S_2^b)', \Mod(S_2^b)']\) is the
+  whole of \(\Mod(S_2^b)'\).
 \end{proposition}
 
 % TODO: Comment on the proof of this?
 \begin{proposition}\label{thm:commutator-normal-gen}
   If \(g \ge 2\) and \(\alpha, \beta \subset S_g\) are simple closed curves
-  with \(\#(\alpha \cap \beta) = 1\) then \(\Mod(S_g)'\) is normally generated
-  by \(\tau_\alpha \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha \tau_\beta^{-1}
-  \in N \normal \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\).
+  with \(\#(\alpha \cap \beta) = 1\) then \(\Mod(S_g)'\) is \emph{normally
+  generated} by \(\tau_\alpha \tau_\beta^{-1}\) -- i.e. if \(\tau_\alpha
+  \tau_\beta^{-1} \in N \normal \Mod(S_g)'\) then \(\Mod(S_g)' \subset N\).
 \end{proposition}
 
 \section{The Birman-Hilden Theorem}
@@ -46,8 +47,18 @@
 
 % TODO: Cite the paper by Artin
 \begin{theorem}[Artin]
-  \(B_n = \langle \sigma_1, \ldots, \sigma_{n - 1} : \sigma_i \sigma_{i + 1}
-  \sigma_i = \sigma_{i + 1} \sigma_i \sigma_{i + 1} \forall i \rangle\).
+  \[
+    B_n =
+    \left\langle
+    \sigma_1, \ldots, \sigma_{n - 1} :
+    \begin{aligned}
+      \sigma_i \sigma_{i+1} \sigma_i & = \sigma_{i+1} \sigma_i \sigma_{i+1}
+      \ \text{for all} \ i, \\
+      \sigma_i \sigma_j & = \sigma_j \sigma_i
+        \ \text{for} \ j \ne i + 1 \ \text{and} \ j \ne i - 1
+    \end{aligned}
+    \right\rangle
+  \]
 \end{theorem}
 
 \begin{corollary}
@@ -102,7 +113,7 @@
 \end{example}
 
 % TODO: Cite the original paper?
-\begin{theorem}[Wajnryb]
+\begin{theorem}[Wajnryb]\label{thm:wajnryb-presentation}
   If \(\alpha_0, \ldots, \alpha_g\) are as in 
   % TODO: Reference the drawing of the curves somewhere
   and \(a_i = \tau_{\alpha_i} \in \Mod(S_g)\) are the Humphreys generators then

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -1,50 +1,59 @@
 \chapter{Low-Dimensional Representations}
 
-\begin{theorem}[Korkmaz \cite{korkmaz}]\label{thm:low-dim-reps-are-trivial}
+Having built a solid understanding of the combinatorics of Dehn twists, we are
+now ready to attack the problem of classifying the representations of
+\(\Mod(S_g)\). Indeed, in light of the Wajnryb presentation, a representation
+\(\rho : \Mod(S_g) \to \GL_n(\mathbb{C})\) is nothing other than a choice of
+\(2g + 1\) matrices \(\rho(\tau_{\alpha_1}), \ldots, \rho(\tau_{\alpha_{2g}})
+\in \GL_n(\mathbb{C})\) satisfying the relations \strong{(i)} to \strong{(v)}
+from Theorem~\ref{thm:wajnryb-presentation}.
+
+Historically, these relations have been exploited by Funar \cite{funar},
+Franks-Handel \cite{franks-handel} and others to establish the triviality of
+low-dimensional representions, culminating Korkmaz' recent classification of
+representations of dimension \(n \le 2 g\) for \(g \ge 3\) \cite{korkmaz}. The
+goal of this chapter is providing a concise account of Korkmaz' results,
+starting by\dots
+
+\begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial}
   Let \(S_g^b\) be the surface of genus \(g \ge 1\) and \(b\) boundary
-  components and \(\rho : \Mod(S_g^b) \to \GL(V)\) be an \(m\)-dimensional
-  linear representation for some \(m < 2 g\). Then the image of \(\rho\) is
-  Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial.
+  components and \(\rho : \Mod(S_g^b) \to \GL_n(\mathbb{C})\) be a linear
+  representation with \(n < 2 g\). Then the image of \(\rho\) is Abelian. In
+  particular, if \(g \ge 3\) then \(\rho\) is trivial.
 \end{theorem}
 
-% TODO: Explain the setup of the proof: induction in m and case analysis on the
-% Jordan form
+Like so many of the results we have encountered so far, the proof of
+Theorem~\ref{thm:low-dim-reps-are-trivial} is elementary in nature: we proceed
+by induction on \(g\) and tedious case analysis. We begin by the base case \(g
+= 2\).
 
-% TODO: Explain this is the base case
 \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case}
-  Let \(\rho : \Mod(S_2^b) \to \GL(V)\) be an \(m\)-dimensional representation,
-  \(m \le 3\). Then the image of \(\rho\) is a quotient of \(\mathbb{Z}/10\).
+  Given \(\rho : \Mod(S_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the
+  image of \(\rho\) is Abelian.
 \end{proposition}
 
-% I don't think it's worth including the whole proof in here: the case analysis
-% is too boring and takes too much space
 \begin{proof}[Sketch of proof]
-  % TODO: You haven't commented on the Abelianization in the case with boundary
-  % It is easy to see that Mod(S_2^b)^ab is a quotient of ℤ/10: the map
-  % Mod(S_2^b)^ab → Mod(S_2)^ab induced by the inclusion morphism is surjective
-  Since \(\Mod(S_2^b)^\ab \cong \mathbb{Z}/10\), it suffices to show
-  \(\rho(\Mod(S_2^b))\) is Abelian, so that \(\rho\) factors through the
-  Abelianization map \(\Mod(S_2^b) \to \Mod(S_2^b)^\ab\). Equivalenty, it
-  suffices to show that \(\rho(\Mod(S_2^b)') = 1\). Given \(\alpha \subset
-  S_2^b\), denote \(L_\alpha = \rho(\tau_\alpha)\). Let \(\alpha_1,
-  \alpha_2, \mu_1, \mu_2, \gamma, \eta_1, \ldots, \eta_{b - 1} \subset
-  S_2^b\) be the curves of the Lickorish generators from
-  Theorem~\ref{thm:lickorish-gens}.
+  Given \(\alpha \subset S_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and
+  denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v =
+  \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \mu_1, \mu_2,
+  \gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the
+  Lickorish generators from Theorem~\ref{thm:lickorish-gens}.
   \begin{center}
     \includegraphics[width=.25\linewidth]{images/lickorish-gens-gen-2.eps}
   \end{center}
 
-  If \(m = 1\) then \(\GL(V) = \mathbb{C}^\times\) is Abelian and hence so is
-  \(\rho(\Mod(S_2^b))\). Now if \(m = 2\) or \(3\), by
+  If \(n = 1\) then \(\rho(\Mod(S_2^b)) \subset \GL_1(\mathbb{C}) =
+  \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by
   Propositon~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1}
   = L_{\mu_1}\), so that \(\tau_{\alpha_1} \tau_{\mu_1}^{-1} \in \ker \rho\)
-  and thus \(\Mod(S_2^b)' \subset \ker \rho\). Given the braid relation
+  and thus \(\Mod(S_2^b)' \subset \ker \rho\) -- i.e. \(\rho(\Mod(S_2^b))\) is
+  Abelian. Given the braid relation
   \begin{equation}\label{eq:braid-rel-induction-basis}
     L_{\alpha_1} L_{\mu_1} L_{\alpha_1} = L_{\mu_1} L_{\alpha_1} L_{\mu_1},
   \end{equation}
   this amounts to showing \(L_{\alpha_1}\) and \(L_{\mu_1}\) commute.
 
-  To that end, we exhausively analyse all of the possible Jordan decompositions
+  To that end, we exhausively analyse all of the possible Jordan forms
   \begin{align*}
     \begin{pmatrix}
       \lambda & 0 \\
@@ -106,25 +115,25 @@
     \end{pmatrix}
     & \quad{\normalfont(9)}
   \end{align*}
-  of \(L_{\mu_2}\) in some basis \(\mathcal{B}\) -- where \(\lambda, \mu,
-  \nu \in \mathbb{C}^\times\) are all distinct.
+  of \(L_{\mu_2}\) -- where \(\lambda, \mu, \nu \in \mathbb{C}^\times\) are all
+  distinct. By changing basis we may assume without loss of generality that the
+  matrix \(L_{\mu_2}\) is exactly its Jordan form, so that \(E_{\mu_2 =
+  \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\).
 
   For cases (1) to (7) we use the change of coordinates principle and the braid
-  relation (\ref{eq:braid-rel-induction-basis}) to show that the matrices of
-  \(L_{\alpha_1}\) and \(L_{\mu_1}\) in the basis \(\mathcal{B}\) lie in
-  some Abelian subgroup of \(\GL_m(\mathbb{C})\), \(m = 2\) or \(3\) -- hence
-  they commute. See \cite[Proposition~5.1]{korkmaz} for further details.
-  For cases (8) and (9) we consider the curve \(\alpha_2\). In these cases,
-  the eigenspace \(V_{L_{\mu_2} = \lambda}\) is \(2\)-dimensional. Since
-  \(L_{\mu_2}\) and \(L_{\alpha_2}\) are conjugate, so is \(V_{L_{\alpha_2}
-  = \lambda}\) -- indeed, conjugate operators have the same Jordan form. Now
-  either \(V_{L_{\mu_2} = \lambda} = V_{L_{\alpha_2} = \lambda}\) or
-  \(V_{L_{\mu_2} = \lambda} \ne V_{L_{\alpha_2} = \lambda}\). We begin by
-  the first case.
-
-  We claim that if \(V_{L_{\mu_2} = \lambda} = V_{L_{\alpha_2} = \lambda}\)
-  then \(V_{L_{\alpha_0} = \lambda}\) is \(\Mod(S_2^b)\)-invariant. Indeed, by
-  change of coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with
+  relation (\ref{eq:braid-rel-induction-basis}) to show that \(L_{\alpha_1}\)
+  and \(L_{\mu_1}\) lie in some Abelian subgroup of \(\GL_n(\mathbb{C})\) --
+  hence they commute. See \cite[Proposition~5.1]{korkmaz} for further details.
+  For cases (8) and (9) we consider the curve \(\alpha_2\). In these cases, the
+  eigenspace \(E_{\mu_2 = \lambda}\) is \(2\)-dimensional. Since \(L_{\mu_2}\)
+  and \(L_{\alpha_2}\) are conjugate, \(E_{\alpha_2 = \lambda}\) is also
+  \(2\)-dimensional -- indeed, conjugate operators have the same Jordan form.
+  Now either \(E_{\mu_2 = \lambda} = E_{\alpha_2 = \lambda}\) or \(E_{\mu_2 =
+  \lambda} \ne E_{\alpha_2 = \lambda}\). We begin by the first case.
+
+  We claim that if \(E_{\mu_2 = \lambda} = E_{\alpha_2 = \lambda}\)
+  then \(E_{\mu_2 = \lambda}\) is \(\Mod(S_2^b)\)-invariant. Indeed, by change
+  of coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with
   \begin{align*}
     f \cdot [\mu_2]      & = [\mu_1]
     &
@@ -152,136 +161,141 @@
   \end{align*}
   and thus
   \begin{align*}
-    V_{L_{\mu_1} = \lambda}
-    = \rho(f) V_{L_{\mu_2} = \lambda}
-    & = \rho(f) V_{L_{\alpha_2} = \lambda}
-    = V_{L_{\alpha_1} = \lambda}
+    E_{\mu_1 = \lambda}
+    = \rho(f) E_{\mu_2 = \lambda}
+    & = \rho(f) E_{\alpha_2 = \lambda}
+    = E_{\alpha_1 = \lambda}
     \\
-    V_{L_{\alpha_1} = \lambda}
-    = \rho(g) V_{L_{\mu_2} = \lambda}
-    & = \rho(g) V_{L_{\alpha_2} = \lambda}
-    = V_{L_\gamma = \lambda}
+    E_{\alpha_1 = \lambda}
+    = \rho(g) E_{\mu_2 = \lambda}
+    & = \rho(g) E_{\alpha_2 = \lambda}
+    = E_{\gamma = \lambda}
     \\
-    V_{L_{\eta_i} = \lambda}
-    = \rho(h_i) V_{L_{\mu_2} = \lambda}
-    & = \rho(h_i) V_{L_{\alpha_2} = \lambda}
-    = V_{L_{\alpha_2} = \lambda}.
+    E_{\eta_i = \lambda}
+    = \rho(h_i) E_{\mu_2 = \lambda}
+    & = \rho(h_i) E_{\alpha_2 = \lambda}
+    = E_{\alpha_2 = \lambda}.
   \end{align*}
-  In other words, \(V_{L_{\alpha_1} = \lambda} = V_{L_{\alpha_2} = \lambda} =
-  V_{L_{\mu_1} = \lambda} = V_{L_{\mu_2} = \lambda} = V_{L_\gamma = \lambda} =
-  V_{L_{\eta_1} = \lambda} = \cdots = V_{L_{\eta_{b - 1}} = \lambda}\) is
+  In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} =
+  E_{\mu_1 = \lambda} = E_{\mu_2 = \lambda} = E_{\gamma = \lambda} =
+  E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}\) is
   invariant under the action of all Lickorish generators.
 
   Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^b) \to
-  \GL(V_{L_{\mu_2} = \lambda})\). By case (2), \(\bar \rho(f) = 1\) for all
-  \(f \in \Mod(S_2^b)'\), for \(\bar \rho(\Mod(S_2^b))\) is Abelian. In other
-  words, the matrix of \(\rho(f)\) in the basis \(\mathcal{B}\) has the form
+  \GL(E_{\mu_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\mu_2 =
+  \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar
+  \rho(f) = 1\) for all \(f \in \Mod(S_2^b)'\), given that \(\bar
+  \rho(\Mod(S_2^b))\) is Abelian. Thus
   \[
+    \rho(\Mod(S_2^b)') \subset
     \begin{pmatrix}
       1 & 0 & * \\
       0 & 1 & * \\
       0 & 0 & *
     \end{pmatrix}
   \]
-  and, in particular, it lies inside the group of upper triangular matrices --
-  a solvalbe subgroup of \(\GL_3(\mathbb{C})\). Now by
-  Proposition~\ref{thm:commutator-is-perfect} we get \(\rho(\Mod(S_2^b)') =
-  1\): any homomorphism from a perfect group to a solvable group is trivial.
+  lies inside the group of upper triangular matrices, a solvalbe subgroup of
+  \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we
+  get \(\rho(\Mod(S_2^b)') = 1\): any homomorphism from a perfect group to a
+  solvable group is trivial.
 
-  Finally, if \(V_{L_{\mu_2} = \lambda} \ne V_{L_{\alpha_2} = \lambda}\) and
+  Finally, if \(E_{\mu_2 = \lambda} \ne E_{\alpha_2 = \lambda}\) and
   the Jordan form of \(L_{\mu_2}\) is given by (8) then
   \[
     0
-    \subsetneq V_{L_{\mu_2} = \lambda} \cap V_{L_{\alpha_2} = \lambda}
-    \subsetneq V_{L_{\mu_2} = \lambda}
+    \subsetneq E_{\mu_2 = \lambda} \cap E_{\alpha_2 = \lambda}
+    \subsetneq E_{\mu_2 = \lambda}
     \subsetneq V
   \]
   is a flag of subspaces invariant under \(L_{\mu_1}\) and \(L_{\alpha_1}\),
   for \(\mu_2\) is disjoint from \(\mu_1 \cup \alpha_1\) and thus
   \([\tau_{\mu_2}, \tau_{\mu_1}] = [\tau_{\mu_2}, \tau_{\alpha_1}] = 1\). In
-  this case we can find a basis \(\mathcal{B}'\) for \(V\) in wich the matrices
-  of \(L_{\mu_1}\) and \(L_{\alpha_1}\) are both upper triangular with
-  \(\lambda\) along the diagonal: take \(\mathcal{B}' = \{v_1, v_2, v_3\}\)
-  with \(v_1 \in V_{L_{\mu_2} = \lambda} \cap V_{L_{\alpha_2} = \lambda}\),
-  \(v_2 \in V_{L_{\mu_2}}\) and adjust \(v_3\) to get the desired diagonal
-  entry. Any such pair of matrices satisfying the braid relation
+  this case we can find a basis for \(\mathbb{C}^3\) with respect to wich the
+  matrices of \(L_{\mu_1}\) and \(L_{\alpha_1}\) are both upper triangular with
+  \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in \mathbb{C}^3\) with
+  \(v_1 \in E_{\mu_2 = \lambda} \cap E_{\alpha_2 = \lambda}\), \(v_2 \in
+  V_{L_{\mu_2}}\) and adjust \(v_3\) to get the desired diagonal entry. Any
+  such pair of matrices satisfying the braid relation
   (\ref{eq:braid-rel-induction-basis}) commute.
 
-  Similarly, if \(L_{\mu_2}\) has Jordan form (9) and \(V_{L_{\mu_2} = \lambda}
-  \ne V_{L_{\alpha_2} = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
-  to conclude \([L_{\mu_1}, L_{\alpha_1}] = 1 \in \GL(V)\) -- again, see
-  \cite[Proposition~5.1]{korkmaz} for further details. We are done.
+  Similarly, if \(L_{\mu_2}\) has Jordan form (9) and \(E_{\mu_2 = \lambda}
+  \ne E_{\alpha_2 = \lambda}\) we use (\ref{eq:braid-rel-induction-basis})
+  to conclude \(L_{\mu_1}\) and \(L_{\alpha_1}\) commute -- again, see
+  \cite[Proposition~5.1]{korkmaz}. We are done.
 \end{proof}
 
-\begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
-  Let \(g \ge 3\), \(b \ge 0\) and \(\rho : \Mod(S_g^b) \to \GL(V)\) be an
-  \(m\)-dimensional representation with \(m < 2g\). 
+We are now ready to establish the triviality of low-dimensional
+representations.
 
-  As promised, we proceed by induction on \(g\). If \(g = 1\) then \(m = 1\)
-  and thus \(\rho(\Mod(S_g^b)) \subset \GL(V) = \mathbb{C}^\times\) is Abelian.
-  The base case \(g = 2\) was also established in
-  Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now suppose \(g \ge
-  3\) and every \(n\)-dimensional representation of \(S_{g - 1}^c\) has Abelian
-  image for \(n < 2(g - 1)\) and \(c \ge 0\). Let us show \(\rho\) has Abelian
-  image.
+\begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}]
+  Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(S_g^b) \to
+  \GL_n(\mathbb{C})\) with \(n < 2g\). As promised, we proceed by induction on
+  \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and
+  \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also
+  established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now
+  suppose \(g \ge 3\) and every \(m\)-dimensional representation of \(S_{g -
+  1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has
+  Abelian image.
 
   Let \(\alpha_1, \ldots, \alpha_g, \mu_1, \ldots, \mu_g, \gamma_1, \ldots,
   \gamma_{g - 1}, \beta_1, \ldots, \beta_{b - 1} \subset S_g^b\) be the curves
   from the Lickorish generators of \(\Mod(S_g^b)\), as in
-  Theorem~\ref{thm:lickorish-gens}. Once again, denote \(L_\alpha =
-  \rho(\tau_\alpha)\) for any closed \(\alpha \subset S_g^b\). Let \(R \cong
-  S_{g - 1}^1\) be the closed subsurface highlighted in the following picture.
+  Theorem~\ref{thm:lickorish-gens}. Once again, let \(L_\alpha =
+  \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of
+  \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(R \cong S_{g -
+  1}^1\) be the closed subsurface highlighted in the following picture.
   \begin{center}
     \includegraphics[width=.5\linewidth]{images/lickorish-gens-korkmaz-proof.eps}
   \end{center}
 
   % TODO: Add more comments on the injectivity of this map?
-  We claim that it suffices to find a \(\Mod(R)\)-invariant\footnote{Here we
-  view $\Mod(R)$ as a subgroup of $\Mod(S_g^b)$ via the inclusion homomorphism
-  $\Mod(R) \to \Mod(S_g^b)$ from Example~\ref{ex:inclusion-morphism}, which can
-  be shown to be injective in this particular case.} subspace \(W \subset V\)
-  of dimension \(n\) with \(2 \le n \le m - 2\). Indeed, in this case \(n < 2(g
-  - 1)\) and \(\dim \mfrac{V}{W} = m - n < 2(g - 1)\). Thus both
-  representations
+  We claim that it suffices to find a \(m\)-dimensional
+  \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of
+  $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from
+  Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in
+  this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le
+  n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim
+  \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both representations
   \begin{align*}
-    \rho_1 : \Mod(R) & \to \GL(W) & \rho_2 : \Mod(R) & \to \GL(\mfrac{V}{W})
+    \rho_1 : \Mod(R) & \to \GL(W) \cong \GL_m(\mathbb{C})
+    &
+    \rho_2 : \Mod(R) & \to \GL(\mfrac{\mathbb{C}^n}{W})
+    \cong \GL_{n - m}(\mathbb{C})
   \end{align*}
   fall into the induction hypotesis -- i.e. \(\rho_i(\Mod(R))\) is
   Abelian. In particular, \(\rho_i(\Mod(R)') = 1\) and we can find some
-  basis for \(V\) under which
+  basis for \(\mathbb{C}^n\) with respect to which
   \[
-    % TODO: Make this prettier: somehow format the block sizes in the equation
     \rho(f) =
     \left(
     \begin{array}{c|c}
-      1 & * \\ \hline
-      0 & 1
+      1_m & *         \\ \hline
+        0 & 1_{n - m}
     \end{array}
     \right)
   \]
-  for any \(f \in \Mod(R)'\) -- where the first and second diaogonal blocks are
-  \(n \times n\) and \((m - n) \times (m - n)\). Since the group of upper
-  triangular matrices is solvable, it follows from
-  Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) annihilates all
-  \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1} \tau_{\mu_1}^{-1} \in \ker
-  \rho\). But recall from Proposition~\ref{thm:commutator-normal-gen} that
-  \(\Mod(S_g^b)'\) is normally generated by \(\tau_{\alpha_1}
-  \tau_{\mu_1}^{-1}\), from which we conclude \(\rho(\Mod(S_g^b)') = 1\), as
-  desired.
-
-  As before, we exhaustively analyse all possible Jordan forms for
+  for all \(f \in \Mod(R)'\) -- where \(1_k\) denotes the \(k \times k\)
+  identity matrix. Since the group of upper triangular matrices is solvable, it
+  follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\)
+  annihilates all of \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1}
+  \tau_{\mu_1}^{-1} \in \ker \rho\). But recall from
+  Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^b)'\) is normally
+  generated by \(\tau_{\alpha_1} \tau_{\mu_1}^{-1}\), from which we conclude
+  \(\rho(\Mod(S_g^b)') = 1\), as desired.
+
+  As before, we exhaustively analyse all possible Jordan forms of
   \(L_{\mu_g}\). First, consider the case where we can find eigenvalues
   \(\lambda_1, \ldots, \lambda_k\) of \(L_{\mu_g}\) such that the sum \(W =
-  \bigoplus_i V_{L_{\mu_g} = \lambda_i}\) of the corresponding eigenspaces has
-  dimension \(n\) with \(2 \le n \le m - 2\). In this case, it suffices to
-  observe that since \(\mu_g\) lies outside of \(R\), each \(V_{L_{\mu_g} =
+  \bigoplus_i E_{\mu_g = \lambda_i}\) of the corresponding eigenspaces has
+  dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to
+  observe that since \(\mu_g\) lies outside of \(R\), each \(E_{\mu_g =
   \lambda_i}\) is \(\Mod(R)\)-invariant: the Lickorish generators
   \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\mu_1}, \ldots,
-  \tau_{\mu_{g - 1}}, \tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of \(R
-  \cong S_{g - 1}^1\) all commute with \(\tau_{\mu_g}\) and thus preserve the
-  eigenspaces of it's action on \(V\).
+  \tau_{\mu_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of
+  \(R \cong S_{g - 1}^1\) all commute with \(\tau_{\mu_g}\) and thus preserve
+  the eigenspaces of its action on \(\mathbb{C}^n\).
 
-  If no sum of the form \(\bigoplus_i V_{L_{\mu_g} = \lambda_i}\) has
+  If no sum of the form \(\bigoplus_i E_{\mu_g = \lambda_i}\) has
   dimension lying between \(2\) and \(m - 2\) there must be at most \(2\)
   distinct eigenvalues and all eigenspaces must be either \(1\)-dimensional or
   \((m - 1)\)-dimensional. Hence then the Jordan form of \(L_{\mu_g}\) has to
@@ -323,29 +337,27 @@
     \end{pmatrix}
     & \quad{\normalfont(4)}
   \end{align*}
-  for \(\lambda \ne \mu\). We now analyze each one of these sporadic cases
+  for \(\lambda \ne \mu\). We analyze each one of these sporadic cases
   individually.
 
   For case (1), we use the change of coordinates principle: each
   \(L_{\alpha_i}, L_{\mu_i}, L_{\gamma_i},  L_{\eta_i}\) is conjugate to
   \(L_{\mu_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^b)\) act
-  on \(V\) as scalar multiplication by \(\lambda\) as well. Hence
-  \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2) \(W = \ker
-  (L_{\alpha_{2 - g}} - \lambda)^2\) is a \(2\)-dimensional
-  \(\Mod(R)\)-invariant subspace.
-
-  For cases (3) and (4) we consider two situations: \(V_{L_{\mu_g} =
-  \lambda} \ne V_{L_{\alpha_g} = \lambda}\) or \(V_{L_{\mu_g} =
-  \lambda} = V_{L_{\alpha_g} = \lambda}\). In the first case, \(W =
-  V_{L_{\mu_g} = \lambda} \cap V_{L_{\alpha_g} = \lambda}\) is a
-  \((m - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since \(L_{\alpha_{2
-  - g}}\) and \(L_{\alpha_g}\) are conjugate and \(\alpha_g\) lies
-  outside of \(R\), both \(V_{L_{\mu_g} = \lambda}\) and
-  \(V_{L_{\alpha_g} = \lambda}\) are \(\Mod(R)\)-invariant \((m -
-  1)\)-dimensional subspaces.
-
-  Finally, we consider the case where \(V_{L_{\mu_g} = \lambda} =
-  V_{L_{\alpha_g} = \lambda}\). In this situation, as in the proof of
+  on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence
+  \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2), \(W = \ker
+  (L_{\mu_g} - \lambda)^2\) is a \(2\)-dimensional \(\Mod(R)\)-invariant
+  subspace.
+
+  For cases (3) and (4) we consider two situations: \(E_{\mu_g = \lambda} \ne
+  E_{\alpha_g = \lambda}\) or \(E_{\mu_g = \lambda} = E_{\alpha_g = \lambda}\).
+  In the first case, \(W = E_{\mu_g = \lambda} \cap E_{\alpha_g = \lambda}\) is
+  a \((m - 2)\)-dimensional \(\Mod(R)\)-invariant subspace: since \(L_{\mu_g}\)
+  and \(L_{\alpha_g}\) are conjugate and \(\alpha_g\) lies outside of \(R\),
+  both \(E_{\mu_g = \lambda}\) and \(E_{\alpha_g = \lambda}\) are
+  \(\Mod(R)\)-invariant \((m - 1)\)-dimensional subspaces.
+
+  Finally, we consider the case where \(E_{\mu_g = \lambda} =
+  E_{\alpha_g = \lambda}\). In this situation, as in the proof of
   Proposition~\ref{thm:low-dim-reps-are-trivial-base-case} it follows from the
   change of coordinates principle that there are \(f_i, g_i, h_i \in
   \Mod(S_g^b)\) with
@@ -364,14 +376,14 @@
   \end{align*}
   and thus
   \[
-    V_{L_{\alpha_1} = \lambda} = \cdots = V_{L_{\alpha_g} = \lambda}
-    = V_{L_{\mu_1} = \lambda} = \cdots = V_{L_{\mu_g} = \lambda}
-    = V_{L_{\gamma_1} = \lambda} = \cdots = V_{L_{\gamma_{g - 1}} = \lambda}
-    = V_{L_{\eta_1} = \lambda} = \cdots = V_{L_{\eta_{b - 1}} = \lambda}.
+    E_{\mu_1 = \lambda} = \cdots = E_{\mu_g = \lambda}
+    = E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda}
+    = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda}
+    = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}.
   \]
 
-  In particular, we can find a basis for \(V\) under which the matrix of all
-  Lickorish generators has the form
+  In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which
+  the matrix of any Lickorish generators has the form
   \[
     \begin{pmatrix}
       \lambda & 0       & \cdots & 0       & *      \\
@@ -382,15 +394,128 @@
     \end{pmatrix}.
   \]
   Since the group of upper triangular matrices is solvable and \(\Mod(S_g^b)\)
-  is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. We are done.
+  is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. This concludes
+  the proof \(\rho(\Mod(S_g^b))\) is Abelian.
+
+  To see that \(\rho(\Mod(S_g^b)) = 1\) for \(g \ge 3\) we note that, since
+  \(\rho(\Mod(S_g^b))\) is Abelian, \(\rho\) factors though the Abelinization
+  map \(\Mod(S_g^b) \to \Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g^b),
+  \Mod(S_g^b)]}\) Now recall from Proposition~\ref{thm:trivial-abelianization}
+  that \(\Mod(S_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\)
+  factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done.
 \end{proof}
 
-\begin{theorem}[Korkmaz \cite{korkmaz}]
-  Let \(g \ge 3\) and \(\rho : \Mod(S_g^b) \to \GL(V)\) be a \(2g\)-dimensional
-  linear representation. Then either \(\rho\) is either trivial or conjugate to
-  the symplectic representation\footnote{Here the map \(\Mod(S_g^b) \to
-  \operatorname{Sp}_{2g}(\mathbb{Z})\) is given by the composition of the
-  inclusion morphism \(\Mod(S_g^b) \to \Mod(S_g)\) with the usual symplect
-  representation \(\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\).}
+Having established the triviality of the low-dimensional representations \(\rho
+: \Mod(S_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand
+the \(2g\)-dimensional reprensentations of \(\Mod(S_g^b)\). We certainly know a
+nontrivial example of such, namely the symplectic representation \(\psi :
+\Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from
+Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be
+\emph{essentially} the only example of a nontrivial \(2g\)-dimensional
+representation in the compact case. More precisely,
+
+\begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic}
+  Let \(g \ge 3\) and \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Then
+  \(\rho\) is either trivial or conjugate to the symplectic
+  representation\footnote{Here the map $\Mod(S_g^b) \to
+  \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the
+  inclusion morphism $\Mod(S_g^b) \to \Mod(S_g)$ with the usual symplect
+  representation $\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.}
   \(\Mod(S_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^b)\).
 \end{theorem}
+
+Unfortunately, the limited scope of these master thesis does not allow us to
+dive into the proof of Theorem~\ref{thm:reps-of-dim-2g-are-symplectic}. The
+heart of this proof lies in a somewhat technical result about representations
+of the product \(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers
+to as \emph{the main lemma}. Namely\dots
+
+\begin{lemma}[Korkmaz' Main Lemma]\label{thm:main-lemma}
+  Given \(i = 1, \ldots, n\), denote by \newline \(a_i = (1, \ldots, 1,
+  \sigma_1, 1, \ldots 1)\) and \(b_i = (1, \ldots, 1, \sigma_2, 1, \ldots, 1)\)
+  the \(n\)-tuples in \(B_3^n\) whose \(i\)-th coordinates are \(\sigma_1\) and
+  \(\sigma_2\), respectively, and with all other coordinates equal to \(1\).
+  Let \(m \ge 2n\) and \(\rho : B_3^n \to \GL_m(\mathbb{C})\) be a
+  representation satisfying:
+  \begin{enumerate}
+    \item The only eigenvalue of \(\rho(a_i)\) is \(1\) and it's eigenspace is
+      \((2g - 1)\)-dimensional.
+    \item The eigenspaces of \(\rho(a_i)\) and \(\rho(b_i)\) associated to the
+      eigenvalue \(1\) do not coincide.
+  \end{enumerate}
+  Then \(\rho\) is conjugate to the representation
+  \begin{align*}
+    B_3^n & \to \GL_m(\mathbb{C}) \\
+    a_i
+    & \mapsto 
+    \left(
+    \begin{array}{c|c|c}
+      1_{2(i-1)} & 0                                            & 0 \\ \hline
+               0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline
+               0 & 0                                            & 1_{m-2i}
+    \end{array}
+    \right) \\
+    b_i
+    & \mapsto 
+    \left(
+    \begin{array}{c|c|c}
+      1_{2(i-1)} & 0                                             & 0 \\ \hline
+               0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline
+                 0 & 0                                             & 1_{m-2i}
+    \end{array}
+    \right),
+  \end{align*}
+  where \(1_k\) denotes the \(k \times k\) identity matrix.
+\end{lemma}
+
+This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice
+that for \(n = g\) and \(m = 2g\) the matrices in
+Lemma~\ref{thm:main-lemma} coincide with the action of
+the Lickrish generators \(\tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\alpha_1},
+\ldots, \tau_{\alpha_g} \in \Mod(S_g^b)\) on \(H_1(S_g, \mathbb{C}) \cong
+\mathbb{C}^{2g}\) -- represented in the standard basis \([\mu_1], \ldots,
+[\mu_g], [\alpha_1], \ldots, [\alpha_g]\) for \(H_1(S_g, \mathbb{C})\).
+\begin{align*}
+  (\tau_{\mu_i})_* & =
+  \left(
+  \begin{array}{c|c|c}
+    1 & 0                                            & 0 \\ \hline
+    0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline
+    0 & 0                                            & 1
+  \end{array}
+  \right) &
+  (\tau_{\alpha_i})_* & =
+  \left(
+  \begin{array}{c|c|c}
+    1 & 0                                             & 0 \\ \hline
+    0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline
+    0 & 0                                             & 1
+  \end{array}
+  \right)
+\end{align*}
+
+Hence by embeding \(B_3^g\) in \(\Mod(S_g^b)\) via
+\begin{align*}
+  B_3^g & \to \Mod(S_g^b)         \\
+  a_i   & \mapsto \tau_{\mu_i}    \\
+  b_i   & \mapsto \tau_{\alpha_i}
+\end{align*}
+we can see that any \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\) in a
+certain class of representation satisfying some technical conditions must be
+conjugate to the symplectic representation \(\Mod(S_g^b) \to
+\operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\).
+
+Korkmaz then goes on to show that such technical conditions are met for any
+nontrivial \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore,
+Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with
+respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots,
+\rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots,
+\rho(\tau_{\eta_{b-1}})\) also agrees with the action of \(\Mod(S_g^b)\) on
+\(H_1(S_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional
+representations.
+
+% TODO: Add some final comments about how the rest of the landscape of
+% representations is generally unknown and how there is a lot to study in here
+Recently, Kasahara also classified the \((2g+1)\)-dimensional representations
+of \(\Mod(S_g^b)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology
+groups \cite{kasahara}.

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -178,13 +178,13 @@
   each corresponding to one of the standard generators
   \begin{align*}
     \begin{pmatrix}
-      1 & -1 \\
-      0 &  1
+      1 & 1 \\
+      0 & 1
     \end{pmatrix}
     &&
     \begin{pmatrix}
-      1 & 0 \\
-      1 & 1
+       1 & 0 \\
+      -1 & 1
     \end{pmatrix}
   \end{align*}
   of \(\operatorname{SL}_2(\mathbb{Z})\).