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

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 $\operatorname{PMod}(\Sigma)$. We use the relations arising from a certain pants decomposition of $\Sigma_g$ 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 $g \gg 0$.