arXiv subjects
Publications and source records attributed to Celal Can Bellek.
Wajnryb gave a finite presentation of the handlebody group $\mathcal{M}(V_g)$ and showed that it is generated by five elements for $g\geq 2$. Using his relations, we show that $\mathcal{M}(V_g)$ is generated by three elements if $g\geq 2$.
Let $S(n)$ be the infinite-type surface with infinite genus and $n \in \mathbb{N}$ ends, all of which are accumulated by genus. The mapping class group of this surface, $\mathrm{Map}(S(n))$, is a Polish group that is not countably generated, but it is countably topologically generated. This paper focuses on finding minimal sets of generators for $\mathrm{Map}(S(n))$. We show that for $n \ge 8$, $\mathrm{Map}(S(n))$ is topologically generated by three elements, and for $n \ge 3$, it is topologically generated by four elements. We also establish a generating set of two elements for the Loch Ness Monster surface $S(1)$, and a generating set of three elements for the Jacob's Ladder surface $S(2)$.
Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each accumulated by genus. Although the mapping class groups of these surfaces are not countably generated,they are Polish groups and hence admit a countable topological generating set. We study minimal topological generating sets for $\mathrm{Map}(S(n))$ consisting entirely of torsion elements, with special attention to involutions. In particular, we prove that $\mathrm{Map}(S(n))$ is topologically generated by four involutions for all $n \geq 16$, and by three involutions for the Loch Ness Monster surface ($n = 1$) and the Jacob's Ladder surface ($n = 2$). We also establish that for even $n \geq 8$, $\mathrm{Map}(S(n))$ is topologically generated by four torsion elements of order $n$. For odd $n \geq 8$, it is topologically generated by three torsion elements of order $n$ and one torsion element of order $n - 1$.
Big mapping class groups are the mapping class groups of infinite-type surfaces, that is, surfaces whose fundamental groups are not finitely generated. While mapping class groups of finite-type surfaces have been extensively studied, the theory of big mapping class groups is a recent and rapidly developing area of research. This thesis provides a systematic introduction to the structure and topological generation of big mapping class groups, emphasizing the key differences from the classical finite-type case. It presents a clear exposition of foundational results concerning their topological and algebraic structure, including known results on finite topological generation for a certain family of infinite-type surfaces.