On power subgroups of Dehn twists in hyperelliptic mapping class groups

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This paper contains two topics, the index of a power subgroup in the mapping class group M(0,2n) of a 2n-punctured sphere and in the hyperelliptic mapping class group Δ(g,0) of an oriented closed surface of genus g. The main tool is a projective representation of M(0,2n) obtained through the Kauffman bracket skein module. For M(0,2n), we prove that the normal closure of the fifth power of a half-twist has infinite index. This is the remaining case of a Masbaum's work. For Δ(g,0), we consider the normal closure of m-th powers of Dehn twists along all symmetric simple closed curves. We show the subgroup has infinite index if m≥ 5 and m≠ 6 for any g≥ 2.