memoire-m2

My M2 Memoire on mapping class groups & their representations

git clone: git://git.pablopie.xyz/memoire-m2
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Commit: e9f6ff13f218bc6725abfd19bdc0f918cc353fa0
Parent: 62bd8937fd39c9b1b174edc22b83997727ee73a2
Author: Pablo <pablo-pie@riseup.net>
Date: Thu, 6 Jun 2024 11:57:33 +0000

Wrote the proof for Korkmaz' Theorem 1

Diffstat

10 files changed, 1289 insertions, 24 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	images/.gitignore	2	2	0
Added	images/Makefile	3	3	0
Added	images/lickorish-gens-gen-2.svg	171	171	0
Added	images/lickorish-gens-korkmaz-proof.svg	323	323	0
Added	images/lickorish-gens.svg	315	315	0
Modified	preamble.tex	3	3	0
Modified	sections/introduction.tex	39	39	0
Modified	sections/presentation.tex	39	36	3
Modified	sections/representations.tex	397	387	10
Modified	sections/twists.tex	21	10	11

diff --git a/images/.gitignore b/images/.gitignore
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diff --git a/images/Makefile b/images/Makefile
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diff --git a/preamble.tex b/preamble.tex
@@ -175,6 +175,9 @@
 % General linear group
 \DeclareMathOperator{\GL}{GL}
 
+% Group Abelianization
+\newcommand{\ab}{{\operatorname{ab}}}
+
 % Group action symbol
 \newcommand{\actson}{\,\rotatebox[origin=c]{-90}{$\circlearrowleft$}\,}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -14,6 +14,40 @@
 \end{definition}
 
 \begin{example}
+  Let \(R \subset S\) be a closed subsurface. Given some \(\phi \in \Homeo^+(R,
+  \partial R)\), we may extend \(\phi\) to \(\tilde{\phi} \in \Homeo^+(S,
+  \partial S)\) by setting \(\tilde{\phi}(p) = p\) for \(p \in S\) outside of
+  \(R\) -- which is well defined since \(\phi\) fixes every point in \(\partial
+  R\). This contruction yields a group homomorphism
+  \begin{align*}
+    \Mod(R) & \to \Mod(S) \\
+     [\phi] & \mapsto [\tilde\phi],
+  \end{align*}
+  known as \emph{the inclusion homomorphism}.
+\end{example}
+
+\begin{example}
+  Let \(\alpha \subset \partial S\) be a boundary component of \(S\) and fix
+  some orientation of \(\alpha\). We refer to the inclusion homomorphism
+  \(\operatorname{cap} : \Mod(S) \to \Mod(S \cup_\alpha (\mathbb{D} \setminus
+  \{0\}))\) as \emph{the capping homomorphism}.
+\end{example}
+
+\begin{proposition}
+  Given some oriented boundary component \(\alpha \subset \partial S\) of
+  \(S\), there is an exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      1 \rar &
+      \langle \tau_\alpha \rangle \rar &
+      \Mod(S) \rar{\operatorname{cap}} &
+      \Mod(S \cup_\alpha (\mathbb{D} \setminus \{0\})) \rar &
+      1
+    \end{tikzcd}
+  \end{center}
+\end{proposition}
+
+\begin{example}\label{ex:inclusion-morphism}
   \(\Mod(S) \actson \{ \vec{[\alpha]} : \alpha \subset S \}\) and \(\Mod(S)
   \actson \{ [\alpha] : \alpha \subset S \}\) via
   \begin{align*}
@@ -22,6 +56,7 @@
   \end{align*}
   for \(f = [\phi] \in \Mod(S)\).
 \end{example}
+
 \begin{example}
   \(\Mod(S) \actson H_k(S, \mathbb{R})\)
 \end{example}
@@ -30,6 +65,10 @@
   The symplectic representation.
 \end{example}
 
+\begin{example}
+  TQFT representations.
+\end{example}
+
 \section{First Computations}
 
 % TODO: Explain the Alexander trick

diff --git a/sections/presentation.tex b/sections/presentation.tex
@@ -1,8 +1,43 @@
-\chapter{Wajnryb's Presentation}
+\chapter{Relations Among Twists \& Wajnryb's Presentation}
 
 \begin{proposition}[Lantern relation]
 \end{proposition}
 
+\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}
+
+% TODO: Explain to get the groups in the low-genus settings we use explicit
+% presentations of Mod(S_g)
+\begin{center}
+  \begin{tabular}{ r|c|l }
+    \(g\) & \(S_g\)          & \(\Mod(S_g)^\ab\) \\
+    \hline
+    \(0\) & \(\mathbb{S}^2\) & \(0\)             \\
+    \(1\) & \(\mathbb{T}\)   & \(\mathbb{Z}/12\) \\
+    \(2\) & \(S_2\)          & \(\mathbb{Z}/10\)
+  \end{tabular}
+\end{center}
+
+% 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 \(\Mod(S_2^b)'\).
+\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 \(\Mod(S_g)' \subset N\).
+\end{proposition}
+
+\section{The Birman-Hilden Theorem}
+
 \begin{definition}
   The \emph{braid group on \(n\) strands} \(B_n\) is the fundamental group
   \(\pi_1(C(\mathbb{D}, n), *)\) of the unordered configuration space
@@ -25,8 +60,6 @@
   - 1})^n\).
 \end{example}
 
-\section{The Birman-Hilden Theorem}
-
 \begin{definition}
   Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{symmetric mapping class group
   of \(S_\ell^1\)} is the subgroup \(\SMod(S_\ell^1) = \{ [\phi] \in

diff --git a/sections/representations.tex b/sections/representations.tex
@@ -1,19 +1,396 @@
 \chapter{Low-Dimensional Representations}
 
-\begin{theorem}[Korkmaz \cite{korkmaz}]
+\begin{theorem}[Korkmaz \cite{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 \(n\)-dimensional
-  linear representation for some \(n < 2 g\). Then the image of \(\rho\) is
+  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.
 \end{theorem}
 
+% TODO: Explain the setup of the proof: induction in m and case analysis on the
+% Jordan form
+
+% 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 \(\mathbb{Z}/10\).
+\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}.
+  \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
+  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
+  \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
+  \begin{align*}
+    \begin{pmatrix}
+      \lambda & 0 \\
+      0       & \mu
+    \end{pmatrix}
+    & \quad{\normalfont(1)}
+    &
+    \begin{pmatrix}
+      \lambda & 0 \\
+      0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(2)}
+    &
+    \begin{pmatrix}
+      \lambda & 1 \\
+      0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(3)}
+    \\
+    \begin{pmatrix}
+      \lambda & 0   & 0   \\
+      0       & \mu & 0   \\
+      0       & 0   & \nu
+    \end{pmatrix}
+    & \quad{\normalfont(4)}
+    &
+    \begin{pmatrix}
+      \lambda & 0       & 0       \\
+      0       & \lambda & 0       \\
+      0       & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(5)}
+    &
+    \begin{pmatrix}
+      \lambda & 0   & 0   \\
+      0       & \mu & 1   \\
+      0       & 0   & \mu
+    \end{pmatrix}
+    & \quad{\normalfont(6)}
+    \\
+    \begin{pmatrix}
+      \lambda & 1       & 0       \\
+      0       & \lambda & 1       \\
+      0       & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(7)}
+    &
+    \begin{pmatrix}
+      \lambda & 0       & 0       \\
+      0       & \lambda & 1       \\
+      0       & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(8)}
+    &
+    \begin{pmatrix}
+      \lambda & 0       & 0   \\
+      0       & \lambda & 0   \\
+      0       & 0       & \mu
+    \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.
+
+  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
+  \begin{align*}
+    f \cdot [\mu_2]      & = [\mu_1]
+    &
+    g \cdot [\mu_2]      & = [\alpha_1]
+    &
+    h_i \cdot [\mu_2]    & = [\mu_2]    \\
+    f \cdot [\alpha_2]   & = [\alpha_1]
+    &
+    g \cdot [\alpha_2]   & = [\gamma]
+    &
+    h_i \cdot [\alpha_2] & = [\eta_i].
+  \end{align*}
+  In particular,
+  \begin{align*}
+    f   \tau_{\mu_2}    f^{-1}   & = \tau_{\mu_1}
+    &
+    g   \tau_{\mu_2}    g^{-1}   & = \tau_{\alpha_1}
+    &
+    h_i \tau_{\mu_2}    h_i^{-1} & = \tau_{\mu_2}     \\
+    f   \tau_{\alpha_2} f^{-1}   & = \tau_{\alpha_1}
+    &
+    g   \tau_{\alpha_2} g^{-1}   & = \tau_{\gamma}
+    &
+    h_i \tau_{\alpha_2} h_i^{-1} & = \tau_{\eta_i}.
+  \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}
+    \\
+    V_{L_{\alpha_1} = \lambda}
+    = \rho(g) V_{L_{\mu_2} = \lambda}
+    & = \rho(g) V_{L_{\alpha_2} = \lambda}
+    = V_{L_\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}.
+  \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
+  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
+  \[
+    \begin{pmatrix}
+      1 & 0 & * \\
+      0 & 1 & * \\
+      0 & 0 & *
+    \end{pmatrix}
+  \]
+  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.
+
+  Finally, if \(V_{L_{\mu_2} = \lambda} \ne V_{L_{\alpha_2} = \lambda}\) and
+  the Jordan form of \(L_{\mu_2}\) is given by (8) then
+  \[
+    0
+    \subsetneq V_{L_{\mu_2} = \lambda} \cap V_{L_{\alpha_2} = \lambda}
+    \subsetneq V_{L_{\mu_2} = \lambda}
+    \subsetneq V
+  \]
+  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
+  (\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.
+\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 \(m < 2g\). 
+
+  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.
+
+  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.
+  \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
+  \begin{align*}
+    \rho_1 : \Mod(R) & \to \GL(W) & \rho_2 : \Mod(R) & \to \GL(\mfrac{V}{W})
+  \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
+  \[
+    % TODO: Make this prettier: somehow format the block sizes in the equation
+    \rho(f) =
+    \left(
+    \begin{array}{c|c}
+      1 & * \\ \hline
+      0 & 1
+    \end{array}
+    \right)
+  \]
+  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
+  \(L_{\mu_g}\). 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 \(V_{L_{\mu_g} =
+  \lambda_i}\) 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 \(V\).
+
+  If no sum of the form \(\bigoplus_i V_{L_{\mu_g} = \lambda_i}\) 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
+  be one of
+  \begin{align*}
+    \begin{pmatrix}
+      \lambda & 0       & 0      & \cdots & 0       & 0       \\
+      0       & \lambda & 0      & \ldots & 0       & 0       \\
+      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
+      0       & 0       & 0      & \cdots & \lambda & 0       \\
+      0       & 0       & 0      & \cdots & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(1)}
+    &
+    \begin{pmatrix}
+      \lambda & 1       & 0      & \cdots & 0       & 0       \\
+      0       & \lambda & 1      & \ldots & 0       & 0       \\
+      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
+      0       & 0       & 0      & \cdots & \lambda & 1       \\
+      0       & 0       & 0      & \cdots & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(2)}
+    \\
+    \begin{pmatrix}
+      \lambda & 0       & 0      & \cdots & 0       & 0       \\
+      0       & \lambda & 0      & \ldots & 0       & 0       \\
+      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
+      0       & 0       & 0      & \cdots & \lambda & 1       \\
+      0       & 0       & 0      & \cdots & 0       & \lambda
+    \end{pmatrix}
+    & \quad{\normalfont(3)}
+    &
+    \begin{pmatrix}
+      \lambda & 0       & 0      & \cdots & 0       & 0       \\
+      0       & \lambda & 0      & \ldots & 0       & 0       \\
+      \vdots  & \vdots  & \vdots & \ddots & \vdots  & \vdots  \\
+      0       & 0       & 0      & \cdots & \lambda & 0       \\
+      0       & 0       & 0      & \cdots & 0       & \mu
+    \end{pmatrix}
+    & \quad{\normalfont(4)}
+  \end{align*}
+  for \(\lambda \ne \mu\). We now 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
+  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
+  \begin{align*}
+    f_i \tau_{\mu_g}    f_i^{-1} & = \tau_{\mu_i}
+    &
+    g_i \tau_{\mu_g}    g_i^{-1} & = \tau_{\alpha_i}
+    &
+    h_i \tau_{\mu_g}    h_i^{-1} & = \tau_{\mu_g}
+    \\
+    f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i}
+    &
+    g_i \tau_{\alpha_g} g_i^{-1} & = \tau_{\gamma_i}
+    &
+    h_i \tau_{\alpha_g} h_i^{-1} & = \tau_{\eta_i}.
+  \end{align*}
+  and thus
+  \[
+    V_{L_{\alpha_1} = \lambda} = \cdots = V_{L_{\alpha_g} = \lambda}
+    = V_{L_{\mu_1} = \lambda} = \cdots = V_{L_{\mu_g} = \lambda}
+    = V_{L_{\gamma_1} = \lambda} = \cdots = V_{L_{\gamma_{g - 1}} = \lambda}
+    = V_{L_{\eta_1} = \lambda} = \cdots = V_{L_{\eta_{b - 1}} = \lambda}.
+  \]
+
+  In particular, we can find a basis for \(V\) under which the matrix of all
+  Lickorish generators has the form
+  \[
+    \begin{pmatrix}
+      \lambda & 0       & \cdots & 0       & *      \\
+      0       & \lambda & \ldots & 0       & *      \\
+      \vdots  & \vdots  & \ddots & \vdots  & \vdots \\
+      0       & 0       & \cdots & \lambda & *      \\
+      0       & 0       & \cdots & 0       & *
+    \end{pmatrix}.
+  \]
+  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.
+\end{proof}
 
 \begin{theorem}[Korkmaz \cite{korkmaz}]
-  Let \(\rho : \Mod(S_g^b) \to \GL(V)\) be a \(2g\)-dimensional linear
-  representation. Then either \(\rho\) is either trivial or conjugate to the
-  symplectic representation\footnote{Here the map \(\Mod(S_g^b)\) 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 \(\Mod(S_g^b)\).
+  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 \(\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\).}
+  \(\Mod(S_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^b)\).
 \end{theorem}

diff --git a/sections/twists.tex b/sections/twists.tex
@@ -24,14 +24,10 @@
   Let \(\alpha \subset S\) be a simple closed curve and \(T_\alpha\) be a
   representative of \(\tau_\alpha \in \Mod(S)\). Then \(\# (T_\alpha^k(\beta)
   \cap \beta) = |k| \cdot \#(\alpha \cap \beta)^2\) for any \(k \in
-  \mathbb{Z}\).
+  \mathbb{Z}\). In particular, if \(\alpha\) is nontrivial then \(\tau_\alpha\)
+  has infinite order.
 \end{proposition}
 
-\begin{corollary}
-  Given some  nontrivial \(\alpha \subset S\), \(\tau_\alpha\) has infinite
-  order.
-\end{corollary}
-
 \begin{fact}
   \(\tau_\alpha = \tau_\beta \iff [\alpha] = [\beta]\).
 \end{fact}
@@ -147,11 +143,14 @@
 \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}]
 \end{proof}
 
-\begin{theorem}[Lickorish generators]
-  % TODO: Cite figure
-  If \(g \ge 2\) then \(\Mod(S_g)\) is generated by \(\tau_{\alpha_0}, \ldots,
-  \tau_{\alpha_{2g}}, \tau_{\mu_1}, \ldots, \tau_{\mu_{g - 2}} \in \Mod(S_g)\)
-  as in figure.
+\begin{theorem}[Lickorish generators]\label{thm:lickorish-gens}
+  If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by \(\tau_{\alpha_1},
+  \ldots, \tau_{\alpha_g}, \tau_{\mu_1}, \ldots, \tau_{\mu_g}, \tau_{\gamma_1},
+  \ldots, \tau_{\gamma_{g - 1}}, \tau_{\eta_1}, \ldots, \tau_{\eta_{b - 1}}\) as
+  in the following diagram.
+  \begin{center}
+    \includegraphics[width=.5\linewidth]{images/lickorish-gens.eps}
+  \end{center}
 \end{theorem}
 
 \begin{corollary}[Humphreys generators]