Generating the Extended Mapping Class Group by Three Involutions

Abstract

We prove that the extended mapping class group, $\mathrm{Mo}{\mathrm{d}}^{\ast }({\mathrm{\Sigma }}_{\mathrm{g}})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\ge 5$. In the presence of punctures, we prove that $\mathrm{Mo}{\mathrm{d}}^{\ast }({\mathrm{\Sigma }}_{\mathrm{g},\mathrm{p}})$ can be generated by three involutions for $g\ge 10$ and $p\ge 6$ (with the exception that for $g\ge 11$, $p$ should be at least $15$).

17 pages, 7 figures. v2: We delete Theorem C (couldn't fix our mistake in the proof), on the other hand we improve the result in Theorem B

References in corpus (6)

• Generating Mapping Class Groups by Involutions
• Torsion Elements in the Mapping Class Group of a Surface
• The extended mapping class group is generated by three symmetries
• Generating mapping class group by two torsion elements
• The Torsion Generating Set Of The Extended Mapping Class Groups In Low Genus Cases
• On minimal generating sets for the mapping class group of a punctured surface