Small Sets of Generators for Handlebody Groups

Tülin Altunöz; Celal Can Bellek; Emir Gül; Mehmetcik Pamuk; Oğuz Yıldız

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arXiv · 2607.01003

Small Sets of Generators for Handlebody Groups

Tülin Altunöz·Celal Can Bellek·Emir Gül·Mehmetcik Pamuk·Oğuz Yıldız

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Abstract

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.

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BibTeX RIS

Tülin Altunöz, Celal Can Bellek, Emir Gül, Mehmetcik Pamuk, Oğuz Yıldız. 2026-07-01. Small Sets of Generators for Handlebody Groups. https://arxiv.org/abs/2607.01003

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