The Mapping Class Group of a Punctured Surface Is Generated by Three Elements
نویسنده
چکیده
Let Σg,p be a closed oriented surface of genus g ≥ 1 with p punctures. Let Mod(Σg,p) be the mapping class group of Σg,p. Wajnryb proved in [Wa] that for p = 0, 1 Mod(Σg,p) is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken as a Dehn twist. For p ≥ 2, We proved that Mod(Σg,p) is generated by three elements.
منابع مشابه
Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups
In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasiisometric rigidity theorem for the mapping class group MCG(S g ) of a once punctured surface S g : if K is a finitely generated group quasi-isometric to MCG(S g ) then there is a homomorphism K → MCG(S g ) with finite kernel and fini...
متن کاملPresentations for the punctured mapping class groups in terms of Artin groups
Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F , and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h : F → F which pointwise fix the boundary of F and such that h(P) = P . In this paper, we calculate presentations for...
متن کامل0 N ov 1 99 9 Presentations for the punctured mapping class groups in terms of Artin groups Catherine
Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F , and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h : F → F which pointwise fix the boundary of F and such that h(P) = P. In this paper, we calculate presentations for ...
متن کاملGenerating the Surface Mapping Class Group by Two Elements
Wajnryb proved in [W2] that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist.
متن کاملGenerating the mapping class group of a punctured surface by involutions
Let Σg,b denote a closed orientable surface of genus g with b punctures and let Mod(Σg,b) denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, Mod(Σg,b) is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate Mod(Σg,b). B...
متن کامل