Geometric representations of the braid groups (Q2811904)

Let $\Sigma_{g,b}$ be the orientable connected compact surface of genus $g$ with $b$ boundary components, $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$ the associated mapping class group globally preserving each boundary component and the mapping class group $\mathcal{M}\mathrm{od}(\Sigma_{g,b}, \partial \Sigma_{g,b})$ preserving the boundary pointwise. A geometric representation is any representation of a group in the mapping class group of some surface $\Sigma_{g,b}$. The aim of this paper is to describe all geometric representations of the braid group $B_n$ with $n \geq 6$ strands in $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$ subject to the only condition that $g \leq n/2$. NEWLINENEWLINENEWLINEA geometric representation is called cyclic if its image is a cyclic group. Two Dehn twists $T_a$ and $T_b$ along two distinct curves $a$ and $b$ satisfy $T_a T_b T_a = T_b T_a T_b$ if and only if the curves $a$ and $b$ meet in one point, whereas they commute if and only if the curves $a$ and $b$ are disjoint. A monodromy representation of $B_n$ is a geometric representation of $B_n$ which sends the standard generators $\tau_1, \tau_2, \dots, \tau_{n-1}$ of $B_n$ to distinct Dehn twists. A monodromy representation of $B_n$ can be characterized by the data of an integer $\varepsilon \in \{ \pm 1 \}$ and an ordered $(n-1)$-tuple of curves $(a_1, a_2, \dots, a_{n-1})$ such that for all $i, j \in \{ 1, 2, \dots, n-1 \}$, the curves $a_i$ and $a_j$ are disjoint when $|i-j| \not= 1$, and intersect in exactly one point when $|i-j|= 1$. So the monodromy representation $\rho$ associated to the pair $((a_1, a_2, \dots, a_{n-1}), \varepsilon)$ is defined by setting $\rho(\tau_i) = T_{a_i}^{\varepsilon}$ for all $i \leq n-1$. NEWLINENEWLINENEWLINELet $n \geq 3$ be an integer, $G$ be any group, $\rho$ a homomorphism from $B_n$ to $G$ and $w$ an element lying in the centralizer of $\rho(B_n)$ in $G$. The transvection $\rho_1$ of $\rho$ with direction $w$ is the homomorphism defined by setting $\rho_1(\tau_i) = \rho(\tau_i) w$ for all $i \leq n-1$. A transvection of a monodromy representation $\rho$ of $B_n$ is thus characterized by a unique triple $((a_1, a_2, \dots, a_{n-1}), \varepsilon, W)$, where $W$ is a mapping class which preserves each curve $a_i$, $i \leq n-1$, and is defined by setting $\rho(\tau_i) = T_{a_i}^{\varepsilon} W$ for all $i \leq n-1$. By $\Sigma(\rho)$ is denoted the tubular neighborhood of the union of the curves $a_i$ where $i$ ranges from 1 to $n-1$. NEWLINENEWLINENEWLINETheorem 1 is the first main result of the present paper says the following. Let $n \geq $ be an integer and $\Sigma_{g,b}$ a surface with $g \leq n/2$ and $b \geq 0$. Then any representation $\rho$ from $B_n$ to $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$ is either cyclic, or is a transvection of a monodromy representation. Moreover, such transvections of monodromy representation exist if and only if $g \geq n/2 - 1$. NEWLINENEWLINEThis result still holds when we consider $\mathcal{M}\mathrm{od}(\Sigma_{g,b}, \partial \Sigma_{g,b})$ instead of $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$. \par It is possible to characterize the faithfulness of the geometric representation of $B_n$. Theorem 3 proves that if $n \geq 6$ and $\Sigma_{g,b}$ a surface with $g \leq n/2$ and $\rho$ be a homomorphism from $B_n$ to $\mathcal{M}\mathrm{od}(\Sigma_{g,b}, \partial \Sigma_{g,b})$ or to $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$, then NEWLINENEWLINE\begin{itemize} \item[(i)] The homomorphism $\rho : B_n \to \mathcal{M}\mathrm{od}(\Sigma_{g,b}, \partial \Sigma_{g,b})$ is injective if and only if it is a transvection of a monodromy homomorphism such that the boundary components of $\Sigma(\rho)$ do not bound any disk in $\Sigma$. \item[(ii)] The homomorphism $\rho : B_n \to \mathcal{PM}\mathrm{od}(\Sigma_{g,b})$ is injective if and only if it is a transvection of a monodromy homomorphism such that boundary components of $\Sigma(\rho)$ do not bound any disk in $\Sigma_{g,b}$ and at least one boundary component of $\Sigma(\rho)$ is not isotopic to any boundary component of $\Sigma_{g,b}$. \end{itemize}NEWLINENEWLINEFor four families of groups: the braid groups $B_n$ for $n \geq 6$, the Artin groups of type $D_n$ for all $n \geq 6$, the mapping class groups $\mathcal{PM}\mathrm{od}(\Sigma_{g,b})$ and $\mathcal{M}\mathrm{od}(\Sigma_{g,b}, \partial \Sigma_{g,b})$, for $g \geq 2$ and $b \geq 0$, the author describes the structure of the sets of the endomorphisms of these groups, their automorphisms and their outer automorphism groups. Also he describes the set of the homomorphisms $B_n \to B_m$ with $m \leq n + 1$ and the set of the homomorphisms between mapping class groups of surfaces (possibly with boundary). Finally, he describes the set of the geometric representations of the Artin groups of type $E_n$ ($n \in \{ 6, 7, 8 \}$).