×

Generating the mapping class group of a punctured surface by involutions. (English) Zbl 1244.57035

It is a well-known consequence of the lantern relations in mapping class groups that mapping class groups of surfaces can be generated by finitely many involutions. The article under review extends work of Kassabov, which shows that the mapping class group of a surface \(S_{g,b}\) of genus \(g\) with \(b\) punctures is generated by four involutions if \(g>7\) or if \(g=7\) and \(b\) is even, five involutions if \(g>5\) or if \(g=5\) and \(b\) is even, and by six involutions if \(g>3\) or if \(g=3\) and \(b\) is even.
The author complements these results by showing that the mapping class group is generated by four involutions if \(g=7\) and \(b\) is odd, and by five involutions if \(g=5\) and \(b\) is odd.

MSC:

57M99 General low-dimensional topology
57M07 Topological methods in group theory
20F38 Other groups related to topology or analysis

References:

[1] T. E. Brendle and B. Farb, Every mapping class group is generated by 3 torsion elements and by 6 involutions, J. Algebra, 278 (2004), 187-198. · Zbl 1051.57019 · doi:10.1016/j.jalgebra.2004.02.019
[2] S. Gervais, A finite presentation of the mapping class group of a punctured surface, Topology, 40 (2001), No. 4, 703-725. · Zbl 0992.57013 · doi:10.1016/S0040-9383(99)00079-8
[3] M. Kassabov, Generating Mapping Class Groups by Involutions. v1 25 Nov, 2003.
[4] N. Lu, On the mapping class groups of the closed orientable surfaces, Topology Proc., 13 (1988), 293-324. · Zbl 0699.57007
[5] J. MacCarthy and A. Papadopoulos, Involutions in surface mapping class groups, Enseign. Math., 33 (1987), 275-290. · Zbl 0655.57005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.