Generating the mapping class group of a nonorientable surface by crosscap transpositions

Abstract

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus g ≥ 7 is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of g + 2 crosscap transpositions generating the mapping class group.

Torsion normal generators of the mapping class group of a non-orientable surface

“…In the case of a closed non-orientable surface N g it is known that M(N g ) is generated by involutions [22]. The author of this paper together with Szepietowski [14] have recently proved that M(N g ) is normally generated by one element of infinite order for g ≥ 7.…”

Simple infinite presentations for the mapping class group of a compact non-orientable surface

“…It is known that M(N g,n ) can not be generated by only either Dehn twists or crosscap slides for g ≥ 2 and n ≥ 0 (see [8,9]). On the other hand, Leśniak-Szepietowski [7] showed that M(N g,n ) can be generated by only crosscap transpositions for g ≥ 7 and n ≥ 0. So we have a natural problem as follows.…”

Simple infinite presentations for the mapping class group of a compact non-orientable surface

“…It is known that M(N g,n ) can not be generated by only either Dehn twists or crosscap slides for g ≥ 2 and n ≥ 0 (see [9,10]). On the other hand, Leśniak-Szepietowski [8] showed that M(N g,n ) can be generated by only crosscap transpositions for g ≥ 7 and n ≥ 0. So we have a natural problem as follows.…”