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Oguz Yildiz

Publications and source records attributed to Oguz Yildiz.

14 recordsLinked to original sources
arXiv2025

Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures $p\geq 8$ the group $\mathrm{Mod}(Σ_{13,p})$ is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.

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arXiv2025

Minimal Generation of Mapping Class Groups: A Survey of the Nonorientable Case

This chapter provides a comprehensive survey of foundational results and recent advances concerning minimal generating sets for the mapping class group of a nonorientable surface, $\mathrm{Mod}(N_{g})$, and its index-two twist subgroup, $\mathcal{T}_{g}$. Although the theory for orientable surfaces is well established, the nonorientable case presents unique challenges due to the presence of crosscaps, thus requiring generators beyond Dehn twists. We show that, for a sufficiently large genus $g$, both $\mathrm{Mod}(N_{g})$ and $\mathcal{T}_{g}$ are generated by two elements, which is the minimum possible number. The survey details various types of generating sets, including those composed of torsions, involutions, and commutators, illustrating the geometric and algebraic interplay. We unify foundational work with modern breakthroughs and extend results to punctured surfaces, $\mathrm{Mod}(N_{g,p})$, providing explicit generators, relations, and proof sketches with an emphasis on geometric intuition.

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arXiv2023

On the Torsion Generators of the Mapping Class Groups

We study torsion generators for the (extended) mapping class group or the extended mapping class group of a closed connected orientable surface of genus g. We show that for every g is grater than or equal to 14, mapping class group can be generated by two torsion elements of order g+1 if g is even, and of orders g+1 and g+1 if g is odd. Also for g grater than or equal to 16, mapping class group can be generated by two torsion elements of orders g+1 if g+1 is not divisible by 3, and of orders g+1 and g+1 if g+1 is divisible by 3. Similarly, we obtain two torsion elements generating extended mapping class groups.

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arXiv2023

Involution Generators of the Big Mapping Class Group

Let S = S(n) denote the infinite surface with n ends, n \in N, accumulated by genus. For n \geq 6, we show that the mapping class group of S is topologically generated by five involutions. When n \geq 3, it is topologically generated by six involutions.

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arXiv2022

On the involution generators of the mapping class group of a punctured surface

Let Mod(Sigma_{g, p}) denote the mapping class group of a connected orientable surface of genus g with p punctures. For every even integer p \geq 10 and g \geq 14, we prove that Mod(Sigma_{g, p}) can be generated by three involutions. If the number of punctures p is odd and \geq 9, we show that Mod(Sigma_{g, p}) for g \geq 13 can be generated by four involutions. Moreover, we show that for an even integer p \geq 4 and 3 \leq g \geq 6, Mod(Sigma_{g, p}) can be generated by four involutions.

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arXiv2021

Generating the Extended Mapping Class Group by Three Involutions

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 $15$).

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arXiv2021

The Twist Subgroup is generated by two elements

We show that the twist subgroup $\mathcal{T}_g$ of a nonorientable surface of genus $g$ can be generated by two elements for every odd $g\geq27$ and even $g\geq42$. Using these generators, we can also show that $\mathcal{T}_g$ can be generated by two or three commutators depending on $g$ modulo $4$. Moreover, we show that $\mathcal{T}_g$ can be generated by three elements if $g\geq 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo $4$ again.

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arXiv2020

Torsion Generators of the Twist Subgroup

We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.

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arXiv2020

Generating mapping class group by two torsion elements

We prove that the mapping class group of a closed connected orientable surface of genus $g$ is generated by two elements of order $g$ for $g\geq 6$. Moreover, for $g\geq 7$ we found a generating set of two elements, of order $g$ and $g'$ which is the least divisor of $g$ such that $g'>2$. Furthermore, we proved that it is generated by two elements of order $g/\gcd (g,k)$ for $g\geq 3k^2+4k+1$ and any positive integer $k$.

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arXiv2020

Generating the Twist Subgroup by Involutions

For a nonorientable surface, the twist subgroup is an index 2 subgroup of the mapping class group. It is generated by Dehn twists about two-sided simple closed curves. In this paper, we study involution generators of the twist subgroup. We give generating sets of involutions with the smallest number of elements our methods allow.

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