arXiv subjects

Emir Gül

Publications and source records attributed to Emir Gül.

3 recordsLinked to original sources

arXiv2026

Small Sets of Generators for Handlebody Groups

The mapping class group of a $3$-dimensional handlebody of genus $g$, denoted by $\mathcal{M}(V_g)$, is a fundamental object of study in geometric topology. Building upon the initial generators introduced by Suzuki and their explicit formulation by Takahashi, Wajnryb established that $\mathcal{M}(V_g)$ is generated by exactly five elements for $g \ge 2$. Motivated by recent minimality results in related subgroups we investigate further reductions to this generating set. Through the use of the relations in Wajnryb's presentation, we show that for $g \geq 5$, the handlebody group $\mathcal{M}(V_g)$ is generated by three elements, and for $g \geq 3$, $\mathcal{M}(V_g)$ is generated by four elements, reducing Wajnryb's generating set of five elements by two and one respectively.

math.GT↗

arXiv2026

Small Sets of Topological Generators for Big Mapping Class Groups

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 $S(2)$.

math.GT↗

arXiv2026

Small Torsion Topological Generators for Big Mapping Class Groups

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 $n - 1$.

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