A NOTE ON THE ABELIANIZATIONS OF FINITE-INDEX SUBGROUPS OF THE MAPPING CLASS GROUP

作者：Putman Andrew^*

来源：Proceedings of the American Mathematical Society, 2010, 138(2): 753-758.

For some g >= 3, let F be a finite index subgroup of the mapping class group of a genus g surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of Gamma should be finite. In the paper we prove two theorems supporting this conjecture. For the first, let T(x) denote the Dehn twist about a simple closed curve x. For some n >= 1, we have T(x)(n) is an element of Gamma. We prove that T(x)(n) is torsion in the abelianization of Gamma. Our second result shows that the abelianization of Gamma is finite if Gamma contains a "large chunk" (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves.

• 出版日期2010-2