Abelian subgroups of the Torelli group

Fuzzy Subgroups of Rank Two Abelian p-Group

In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the s...

The Torelli group and congruence subgroups of the mapping class group

Let Σg,n be a compact oriented genus g surface with n boundary components. The mapping class group of Σg,n, denoted Modg,n, is the group of orientation-preserving diffeomorphisms of Σg,n that restrict to the identity on ∂Σg,n, modulo isotopies that fix ∂Σg,n. The group Modg,n plays a fundamental role in many areas of mathematics, ranging from low-dimensional topology to algebraic geometry. At l...

The number of Fuzzy subgroups of some non-abelian groups

In this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. Explicit formulas are givenfor dihedral groups $D_{2n}$, quasi-dihedral groups $QD_{2^n}$, generalized quaternion groups $Q_{4n}$ and modular $p$-groups $M_{p^n}$.

Finite abelian subgroups of the Cremona group of the plane

Generating the Torelli Group

We give a new proof of the theorem of Birman–Powell that the Torelli subgroup of the mapping class group of a closed orientable surface of genus at least 3 is generated by simple homeomorphisms known as bounding pair maps. The key ingredient is a proof that the subcomplex of the curve complex of the surface spanned by curves within a fixed homology class is connected.