The extended mapping class group is generated by three symmetries
英文摘要:
We prove that the extended mapping class group is generated by three orientation reversing involutions.
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Michal Stukow
We prove that the extended mapping class group is generated by three orientation reversing involutions.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configurati…
An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem.
An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the s…
We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.
We prove that the extended mapping class group, $\rm Mod^{*}(Σ_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\rm Mod^{*}(Σ_{g,p})$ can be generated by three involutions for $g\geq 10$ and $p\geq 6$ (with the exception that for $g\geq 11$, $p$ should be at least $15$).
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group…
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping class group is generated by three orientation reversing involutions.
We prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group. In order to apply our theorem to a normal subgroup we require that the "minimal supports" of its elements satisfy a certain complexity condition that is easy to check in practice. The key ingredient is proving that the automorphism groups of many simplicial complexes associated to punctured surfaces are isomorphic to the extended mapping class group. This r…
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Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements. One of these generators is an order 2 element, and the other one is an order 4g+2 element.
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