Glasnik Matematicki, Vol. 49, No. 2 (2014), 337-350.

ON FINITE INDEX SUBGROUPS OF THE MAPPING CLASS GROUP OF A NONORIENTABLE SURFACE

Błażej Szepietowski

Abstract.   Let M(N_h,n) denote the mapping class group of a compact nonorientable surface of genus h ≥ 7 and n ≤ 1 boundary components, and let T(N_h,n) be the subgroup of M(N_h,n) generated by all Dehn twists. It is known that T(N_h,n) is the unique subgroup of M(N_h,n) of index 2. We prove that T(N_h,n) (and also M(N_h,n)) contains a unique subgroup of index 2^g-1(2^g-1) up to conjugation, and a unique subgroup of index 2^g-1(2^g+1) up to conjugation, where g = ⌊(h-1)/2⌋. The other proper subgroups of T(N_h,n) and M(N_h,n) have index greater than 2^g-1(2^g+1). In particular, the minimum index of a proper subgroup of T(N_h,n) is 2^g-1(2^g-1).

2010 Mathematics Subject Classification.   20F38, 57N05.

Key words and phrases.   Mapping class group, nonorinatble surface, finite index subgroup.

DOI: 10.3336/gm.49.2.08

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