منابع مشابه
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.
متن کاملThe 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...
متن کاملAn infinite presentation of the Torelli group
In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious o...
متن کاملCutting and pasting in the Torelli group
We introduce machinery to allow “cut-and-paste”-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a category TSur whose objects are surfaces together with a decoration restricting how th...
متن کاملSmall generating sets for the Torelli group
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup Ig of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called the handle graph on which Ig acts cocompactly.
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ژورنال
عنوان ژورنال: Kodai Mathematical Journal
سال: 2003
ISSN: 0386-5991
DOI: 10.2996/kmj/1073670611