Every mapping class group is generated by 6 involutions

By Tara E. Brendle, Benson Farb. 2003

48 citations recorded by the source corpus.

Abstract Let Mod g , b denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Torsion elements in the mapping class group of a surface, math.GT/0004048 , v1 8Apr2000] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod g , b . We answer Luo's question by proving that 3 torsion elements suffice to generate Mod g ,0 . We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod g , b but also for the order of those elements. In particular, our main result is that 6 involutions (i.e., orientation-preserving diffeomorphisms of order two) suffice to generate Mod g , b for every genus g ⩾3, b =0 and g ⩾4, b =1.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c767b57ff2925c475a6ba51e3ed2e52cfa4b401

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Citations in the corpus

Some problems on mapping class groups and moduli space (2006, methodology): “eldy when g>3. One approach to Problem 5.7 is to follow the original plan of [Jo2], but using a simpler generating set for Mod g. This was indeed the motivation for Brendle and me when we found in [BFa1] a generating set for Mod g consisting of 6 involutions, i.e. 6 elements of order 2. This boun”
Generating the surface mapping class group by two elements (2003, background): “Recently, it was shown by Brendle and Farb in [1] that the mapping class group is generated by three torsion elements and by six involutions.”