By the Birman-Hilden theory, the braid group on 2g strands is embedded in the mapping class group of the closed surface of genus 2g possibly with marked points.
In this talk, using some right-anlged Artin groups in the mapping class groups, we show that any finite index subgroup of the braid group on 2g+1 strands cannot be embedded in the mapping class group of the closed surface of genus g with at most one marked point.