#### Title

Normalizers and Centralizers of Elementary Abelian Subgroups of the Mapping Class Group

#### Document Type

Article

#### Publication Date

1-1-1992

#### Abstract

Let *F* be a finite subgroup of the mapping class group of a closed surface S of genus greater than 2, and let *N(F)* be the normalizer of *F* in the mapping class group. The main result of this paper is to give a description of *N(F)* as an extension of *F* by an explicitly defined subgroup of finite index in the mapping class group of an associated quotient surface *S/F* , punctured at the branch points. The quotient group *N(F)/F* fits into an exact sequence* N _{B} -> N (F)/F -> N_{S}* , where

*N*is of finite index in a braid group and

_{B}*N*is of finite index in the mapping class group of

_{S}*S/F*. A characterization of the image of

*N(F)*in

*Aut(F)*is given so that the centralizer,

*Z(F*), is also determined. These results are then applied to the important case of elementary abelian

*p*-subgroups of mapping class groups. Sample results include a classification of all elementary abelian subgroups with finite normalizers and a description of the normalizers of the rank 1 elementary abelian subgroups.

#### Recommended Citation

Broughton, S.A. (1992). Normalizers and centralizers of elementary abelian subgroups of the mapping class group. In Topology '90, 77-90.