Lower bounds for faithful linear representations of subgroups of the mapping class group

TL;DR

The paper tackles lower bounds on the dimension of faithful linear representations of subgroups of the mapping class group, notably the pure mapping class group $\mathrm{PMod}(\Sigma_g)$. It introduces a refined analysis of commutation relations via a pants-decomposition-derived curve system and studies the associated matrix relations among $M_i$ and $N_j$ derived from Dehn twists. The main result shows that for genus $g \ge 7$ and dimension $d \le 4g-4$, the subgroup $\mathrm{SIP}_0(\Sigma)$ lies in the kernel, forcing the global bound $d(\Sigma) \ge 4g-3$, with analogous lower bounds for Johnson-related groups and pure braid groups. The work provides new techniques based on simple intersection maps and free-group quotients that illuminate the structure of linear representations of Torelli-related subgroups and their impact on broader questions of linearity in low-dimensional topology.

Abstract

Recently, Korkmaz established the lower bound of $3g - 2$ for the dimension of a faithful representation of the mapping class group of an orientable surface of genus $g \ge 3$. We raise this bound to $4g - 3$ in the setting of surfaces of genus $g \ge 7$. A new ingredient is a finer study of the commutation relations in $\operatorname{PMod}(Σ)$. We use the relations arising from a certain pants decomposition of $Σ_g$ to show that any representation of dimension $\le 4g - 4$ is forced to kill a natural subgroup of the Torelli group. We also establish lower bounds for the dimension of faithful representations of related groups: the Johnson group of a closed surface, arbitrarily low terms of the Johnson filtration of a compact surface with one boundary component, and pure braid groups. These lower bounds grow linearly on the genus of the surfaces and the number of strands of the braids. Finally, we also provide some evidence that greater lower bounds for the low-genus cases should lead to improved lower bounds for $g \gg 0$.

Figures (8)

• Figure 1: A basis for the homology of $\Sigma_g$.
• Figure 2: The graph $\Delta_n$: the vertices $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ form two disjoint $n$-cliques, while the vertices $a_i$ and $b_j$ are connected by an edge if and only if $i \ne j$.
• Figure 3: The curves $a_1, \ldots, a_{g-1}, c_1, \ldots, c_{g-1} \subseteq \Sigma$.
• Figure 4: The curves $a_1, \ldots, a_k, b_1, \ldots, b_k \subseteq \Sigma_g^1$.
• Figure 5: The subsurfaces $S_1, \ldots, S_n \subseteq \Sigma_g^1$.

Theorems & Definitions (47)

• Theorem 1: Theorem \ref{['thm:bound-on-faithful-mcg-real']}
• Remark
• Theorem 2: Corollary \ref{['thm:small-dim-rep-kills-separating-twists']}
• Theorem 3: Corollary \ref{['thm:bound-on-faithful-johnson-filtration-real']}
• Theorem 4: Corollary \ref{['thm:bound-on-faithful-braid-real']}
• Remark
• Theorem 5: Theorem \ref{['thm:bigger-bounds-real']}
• Remark
• Theorem 2.1: Dehn--Lickorish, dehn-38lickorish-62lickorish-64
• Theorem 2.2: Powell, powell-78