- Commit
- e9f6ff13f218bc6725abfd19bdc0f918cc353fa0
- Parent
- 62bd8937fd39c9b1b174edc22b83997727ee73a2
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Wrote the proof for Korkmaz' Theorem 1
memoire-m2
My M2 Memoire on mapping class groups & their representations
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 @@ -0,0 +1,2 @@ +*.eps +*-converted-to.pdf
diff --git a/images/Makefile b/images/Makefile @@ -0,0 +1,3 @@ + +%.eps: %.svg + inkscape -o $@ $^
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diff --git a/preamble.tex b/preamble.tex @@ -175,6 +175,9 @@ % General linear group \DeclareMathOperator{\GL}{GL} +% Group Abelianization +\newcommand{\ab}{{\operatorname{ab}}} + % Group action symbol \newcommand{\actson}{\,\rotatebox[origin=c]{-90}{$\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]