memoire-m2
My M2 Memoire on mapping class groups & their representations
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\chapter{Low-Dimensional Representations}\label{ch:representations} Having built a solid understanding of the combinatorics of Dehn twists, we are now ready to attack the problem of classifying the representations of \(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy is to make use of the \emph{geometrically-motivated} relations derived in Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations}. Historically, these relations have been exploited by Funar \cite{funar}, Franks-Handel \cite{franks-handel} and others to establish the triviality of low-dimensional representations, culminating in Korkmaz' \cite{korkmaz} recent classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\). The goal of this chapter is to provide a concise account of Korkmaz' results. \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\) boundary components and \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) 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} Like some 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 \(g = 2\). \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case} Given \(\rho : \Mod(\Sigma_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the image of \(\rho\) is Abelian. \end{proposition} \begin{proof}[Sketch of proof] Given \(\alpha \subset \Sigma_2^b\), 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 \(\alpha_1, \alpha_2, \beta_1, \beta_2, \gamma, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_2^b\) be the curves of the Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in Figure~\ref{fig:lickorish-gens-genus-2}. \begin{figure} \centering \includegraphics[width=.2\linewidth]{images/lickorish-gens-gen-2.eps} \caption{The Lickorish generators for $g = 2$.} \label{fig:lickorish-gens-genus-2} \end{figure} If \(n = 1\) then \(\rho(\Mod(\Sigma_2^b)) \subset \GL_1(\mathbb{C}) = \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1} = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker \rho\) and thus \(\Mod(\Sigma_2^b)' \subset \ker \rho\) -- i.e. \(\rho(\Mod(\Sigma_2^b))\) is Abelian. Given the braid relation \begin{equation}\label{eq:braid-rel-induction-basis} L_{\alpha_1} L_{\beta_1} L_{\alpha_1} = L_{\beta_1} L_{\alpha_1} L_{\beta_1}, \end{equation} this amounts to showing \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute. To that end, we exhaustively analyze all of the possible Jordan forms \begin{align*} \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} & \quad{\normalfont(1)} & \begin{pmatrix} \lambda & 0 \\ 0 & \mu \end{pmatrix} & \quad{\normalfont(2)} & \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} & \quad{\normalfont(3)} \\ \begin{pmatrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{pmatrix} & \quad{\normalfont(4)} & \begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & \nu \end{pmatrix} & \quad{\normalfont(5)} & \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix} & \quad{\normalfont(6)} \\ \begin{pmatrix} \lambda & 0 & 0 \\ 0 & \mu & 1 \\ 0 & 0 & \mu \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 \(L_{\alpha_2}\) -- where \(\lambda, \mu, \nu \in \mathbb{C}^\times\) are all distinct. By changing basis we may assume without loss of generality that the matrix \(L_{\alpha_2}\) is exactly its Jordan form, so that \(E_{\alpha_2 = \lambda} = \bigoplus_{i \le \dim E_{\alpha_2}} \mathbb{C} e_i\). For cases (1) to (6), we use the change of coordinates principle and different relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) lie inside some Abelian subgroup of \(\GL_n(\mathbb{C})\). \begin{enumerate}[leftmargin=1.9cm] \item[\bfseries\color{highlight}(1) \& (4)] By the change of coordinates principle, both \(L_{\alpha_1}\) and \(L_{\beta_1}\) are conjugate to \(L_{\alpha_2} = \lambda\). But the only matrix conjugate to \(\lambda\) is \(\lambda\) itself. Hence \(L_{\alpha_1} = L_{\beta_1} = \lambda \in \mathbb{C}^\times\). \item[\bfseries\color{highlight}(2) \& (5)] Since \(\alpha_2\) is disjoint from both \(\alpha_1\) and \(\beta_1\), it follows from the disjointness relations \([\tau_{\alpha_1}, \tau_{\alpha_2}] = [\tau_{\beta_1}, \tau_{\alpha_2}] = 1\) that \(L_{\alpha_1}\) and \(L_{\beta_1}\) preserve the eigenspaces of \(L_{\alpha_2}\), which are all \(1\)-dimensional. Hence \(L_{\alpha_1}\) and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices -- an Abelian subgroup of \(\GL_n(\mathbb{C})\). \item[\bfseries\color{highlight}(3) \& (6)] As before, it follows from the disjointness relations that \(E_{\alpha_2 = \lambda} = \ker (L_{\alpha_2} - \lambda)\) and \(\ker (L_{\alpha_2} - \lambda)^2\) are invariant under both \(L_{\alpha_1}\) and \(L_{\beta_1}\). This implies \(L_{\alpha_1}\) and \(L_{\beta_1}\) are upper triangular matrices with \(\lambda\) along their diagonals. Any such pair of matrices satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute. \end{enumerate} Similarly, in case (7) we use the braid relation and the disjointness relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute -- see \cite[Proposition~5.1]{korkmaz} for a full proof. Cases (8) and (9) require some extra thought. We consider the curve \(\beta_2\). In these cases, the eigenspace \(E_{\alpha_2 = \lambda}\) is \(2\)-dimensional. Since \(L_{\alpha_2}\) and \(L_{\beta_2}\) are conjugate, \(E_{\beta_2 = \lambda}\) is also \(2\)-dimensional -- indeed, conjugate operators have the same Jordan form. Now either \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) or \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\). We begin by the first case. We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^b)\)-invariant. Indeed, by Observation~\ref{ex:change-of-coordinates-crossing} we can always find \(f, g, h_i \in \Mod(\Sigma_2^b)\) with \begin{align*} f \cdot [\alpha_2] & = [\alpha_1] & g \cdot [\alpha_2] & = [\beta_1] & h_i \cdot [\alpha_2] & = [\alpha_2] \\ f \cdot [\beta_2] & = [\beta_1] & g \cdot [\beta_2] & = [\gamma] & h_i \cdot [\beta_2] & = [\eta_i]. \end{align*} In particular, \begin{align*} f \tau_{\alpha_2} f^{-1} & = \tau_{\alpha_1} & g \tau_{\alpha_2} g^{-1} & = \tau_{\beta_1} & h_i \tau_{\alpha_2} h_i^{-1} & = \tau_{\alpha_2} \\ f \tau_{\beta_2} f^{-1} & = \tau_{\beta_1} & g \tau_{\beta_2} g^{-1} & = \tau_{\gamma} & h_i \tau_{\beta_2} h_i^{-1} & = \tau_{\eta_i}. \end{align*} and thus \begin{align*} E_{\alpha_1 = \lambda} = \rho(f) E_{\alpha_2 = \lambda} & = \rho(f) E_{\beta_2 = \lambda} = E_{\beta_1 = \lambda} \\ E_{\beta_1 = \lambda} = \rho(g) E_{\alpha_2 = \lambda} & = \rho(g) E_{\beta_2 = \lambda} = E_{\gamma = \lambda} \\ E_{\eta_i = \lambda} = \rho(h_i) E_{\alpha_2 = \lambda} & = \rho(h_i) E_{\beta_2 = \lambda} = E_{\beta_2 = \lambda}. \end{align*} In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} = E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\) is invariant under the action of all Lickorish generators. Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(\Sigma_2^b) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^b)'\), given that \(\bar \rho(\Mod(\Sigma_2^b))\) is Abelian. Thus \[ \rho(\Mod(\Sigma_2^b)') \subset \begin{pmatrix} 1 & 0 & * \\ 0 & 1 & * \\ 0 & 0 & * \end{pmatrix} \] lies inside the group of upper triangular matrices, a solvable subgroup of \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we get \(\rho(\Mod(\Sigma_2^b)') = 1\): any homomorphism from a perfect group to a solvable group is trivial. Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and the Jordan form of \(L_{\alpha_2}\) is given by (8) then the disjointness relations \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2}, \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2}, \tau_{\beta_1}] = 1\) implies that \(L_{\alpha_1}\) and \(L_{\beta_1}\) preserve the eigenspaces of both \(L_{\alpha_2}\) and \(L_{\beta_2}\), so \[ 0 \subsetneq E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda} \subsetneq E_{\alpha_2 = \lambda} \subsetneq V \] is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\). In this case we can find a basis for \(\mathbb{C}^3\) with respect to which the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are both upper triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda}\) and \(v_2 \in V_{L_{\alpha_2}}\). Any such pair of matrices satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute. Similarly, if \(L_{\alpha_2}\) has Jordan form (9) and \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) we use (\ref{eq:braid-rel-induction-basis}) to conclude \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute -- again, see \cite[Proposition~5.1]{korkmaz}. We are done. \end{proof} We are now ready to establish the triviality of low-dimensional representations. \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}] Let \(g \ge 1\), \(b \ge 0\), and fix \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) with \(n < 2g\). We want to show \(\rho(\Mod(\Sigma_g^b))\) is Abelian. 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 \(\Sigma_{g - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_g^b\) be the curves from the Lickorish generators of \(\Mod(\Sigma_g^b)\), as in Figure~\ref{fig: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 \(\Sigma \cong \Sigma_{g - 1}^1\) be the closed subsurface highlighted in Figure~\ref{fig:korkmaz-proof-subsurface}. \begin{figure}[ht] \centering \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps} \caption{The subsurface $\Sigma \subset \Sigma_g^b$.} \label{fig:korkmaz-proof-subsurface} \end{figure} We claim that it suffices to find a \(m\)-dimensional \(\Mod(\Sigma)\)-invariant\footnote{Here we view $\Mod(\Sigma)$ as a subgroup of $\Mod(\Sigma_g^b)$ via the inclusion homomorphism $\Mod(\Sigma) \to \Mod(\Sigma_g^b)$ 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(\Sigma) & \to \GL(W) \cong \GL_m(\mathbb{C}) & \rho_2 : \Mod(\Sigma) & \to \GL(\mfrac{\mathbb{C}^n}{W}) \cong \GL_{n - m}(\mathbb{C}) \end{align*} fall into the induction hypothesis -- i.e. \(\rho_i(\Mod(\Sigma))\) is Abelian. In particular, \(\rho_i(\Mod(\Sigma)') = 1\) and we can find some basis for \(\mathbb{C}^n\) with respect to which \[ \rho(f) = \left( \begin{array}{c|c} 1_m & * \\ \hline 0 & 1_{n - m} \end{array} \right) \] for all \(f \in \Mod(\Sigma)'\) -- 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 \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker \rho\). Now recall from Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired. As before, we exhaustively analyze all possible Jordan forms of \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues \(\lambda_1, \ldots, \lambda_k\) of \(L_{\alpha_g}\) such that the sum \(W = \bigoplus_i E_{\alpha_g = \lambda_i}\) of the corresponding eigenspaces has dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to observe that since \(\alpha_g\) lies outside of \(\Sigma\), each \(E_{\alpha_g = \lambda_i}\) is \(\Mod(\Sigma)\)-invariant: the Lickorish generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1}, \ldots, \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 2}}\) of \(\Sigma \cong \Sigma_{g - 1}^1\) all commute with \(\tau_{\alpha_g}\) and thus preserve the eigenspaces of its action on \(\mathbb{C}^n\). If no sum of the form \(\bigoplus_i E_{\alpha_g = \lambda_i}\) has dimension lying between \(2\) and \(n - 2\), then there must be at most \(2\) distinct eigenvalues \(\lambda\) of \(L_{\alpha_g}\), and \(\dim E_{\alpha_g = \lambda} = 1, n - 1, n\) for all such \(\lambda\). Hence the Jordan form of \(L_{\alpha_g}\) 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 analyze the first two sporadic cases individually. \begin{enumerate} \item[\bfseries\color{highlight}(1)] Here we use the change of coordinates principle: each \(L_{\alpha_i}, L_{\beta_i}, L_{\gamma_i}, L_{\eta_i}\) is conjugate to \(L_{\alpha_g} = \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^b)\) act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence \(\rho(\Mod(\Sigma_g^b)) = \langle \lambda \rangle\) is Abelian. \item[\bfseries\color{highlight}(2)] In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a \(2\)-dimensional \(\Mod(\Sigma)\)-invariant subspace. \end{enumerate} For cases (3) and (4), we consider two situations: \(E_{\alpha_g = \lambda} \ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g = \lambda}\). If \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\), then \(W = E_{\alpha_g = \lambda} \cap E_{\beta_g = \lambda}\) is a \((n - 2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(\beta_g\) lies outside of \(\Sigma\) and \(L_{\alpha_g}, L_{\beta_g}\) are conjugate, both \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are \(\Mod(\Sigma)\)-invariant \((n - 1)\)-dimensional subspaces. Finally, we consider the case where \(E_{\alpha_g = \lambda} = E_{\beta_g = \lambda}\). In this situation, as in the proof of Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from Observation~\ref{ex:change-of-coordinates-crossing} that there are \(f_i, g_i, h_i \in \Mod(\Sigma_g^b)\) with \begin{align*} f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i} & g_i \tau_{\alpha_g} g_i^{-1} & = \tau_{\beta_i} & h_i \tau_{\alpha_g} h_i^{-1} & = \tau_{\alpha_g} \\ f_i \tau_{\beta_g} f_i^{-1} & = \tau_{\beta_i} & g_i \tau_{\beta_g} g_i^{-1} & = \tau_{\gamma_i} & h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i} \end{align*} and thus \(E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 = \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda} = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\). In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which the matrix of any Lickorish generator 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 \(\Mod(\Sigma_g^b)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^b))\) is trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^b))\) is Abelian. To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that, since \(\rho\) has Abelian image and thus factors though the Abelianization map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab = \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b), \Mod(\Sigma_g^b)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} that \(\Mod(\Sigma_g^b)^\ab = 0\) for \(g \ge 3\). We are done. \end{proof} Having established the triviality of the low-dimensional representations \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We certainly know a nontrivial example of such, namely the symplectic representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from Example~\ref{ex:symplectic-rep}. Surprisingly, 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 \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Then \(\rho\) is either trivial or conjugate to the symplectic representation\footnote{Here the map $\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual symplectic representation $\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\). \end{theorem} Unfortunately, the limited scope of these master's 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 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}. \begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma} Given \(i = 1, \ldots, n\), denote by \begin{align*} a_i & = (1, \ldots, 1, \sigma_1, 1, \ldots, 1) & b_i & = (1, \ldots, 1, \sigma_2, 1, \ldots, 1) \end{align*} 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 its eigenspace is \((m - 1)\)-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 Lickorish generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^b)\) on \(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \(H_1(\Sigma_g, \mathbb{C})\). \begin{align*} (\tau_{\alpha_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_{\beta_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 embedding \(B_3^g\) in \(\Mod(\Sigma_g^b)\) via \begin{align*} B_3^g & \to \Mod(\Sigma_g^b) \\ a_i & \mapsto \tau_{\alpha_i} \\ b_i & \mapsto \tau_{\beta_i} \end{align*} we can see that any \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\) in a certain class of representation satisfying some technical conditions must be conjugate to the symplectic representation \(\Mod(\Sigma_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\). Korkmaz then goes on to show that such technical conditions are met for any nontrivial \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). 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 agree with the action of \(\Mod(\Sigma_g^b)\) on \(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional representations. Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology groups. On the other hand, the representations of dimension \(n > 2g + 1\) are still poorly understood, and fundamental questions remain unsweared. In the short and mid-terms, the works of Korkmaz and Kasahara lead to many follow-up questions. For example, \begin{enumerate} \item In the \(g \ge 3\) case, Korkmaz \cite[Theorem~3]{korkmaz} established the lower bound of \(3g - 3\) for the dimension of an injective linear representation of \(\Mod(\Sigma_g)\) -- if one such representation exists. Can we improve this lower bound? \item What is the minimal dimension for a representation of \(\Mod(\Sigma_g)\) which does not annihilate the entire kernel of the symplectic representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\)? In particular, do the \((2g + 1)\)-dimensional representations classified by Kasahara \cite{kasahara} annihilate all of \(\ker \psi\)? \end{enumerate} These are some of the questions which I plan to work on during my upcoming PhD.