Handlebodies, Outer space, and tropical geometry
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arXiv
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2025
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| _version_ | 1866910164233224192 |
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| author | Ramadas, Rohini Silversmith, Rob Vogtmann, Karen Winarski, Rebecca R. |
| author_facet | Ramadas, Rohini Silversmith, Rob Vogtmann, Karen Winarski, Rebecca R. |
| contents | The moduli space of graphs $M_{g,n}^{\mathrm{trop}}$ is a polyhedral object that mimics the behavior of the moduli spaces $M_{g,n}$, $\overline{M}_{g,n}$ of (stable) Riemann surfaces; this relationship has been made precise in several different ways, which collectively identify $M_{g,n}^{\mathrm{trop}}$ as the "tropicalization" of $M_{g,n}$. We describe how this relationship lifts to some objects that live over $M_{g,n}$ (like Teichmüller space) and that live over $M_{g,n}^{\mathrm{trop}}$ (like the Culler-Vogtmann space $CV_{g,n}^*$). We introduce the notion of a stable complex handlebody, and show that $CV_{g,n}^*$ can be viewed as the tropicalization of a certain complex manifold $hT(V_{g,n})$ that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification $\overline{hT}(V_{g,n})\supset hT(V_{g,n})$, which we prove is a simply connected complex manifold with simple normal crossings boundary. When $n=0$, $hT(V_{g,n})$ coincides with the moduli space of Schottky groups, $\overline{hT}(V_{g,n})$ coincides with Gerritzen-Herrlich's extended Schottky space, and $CV_{g,0}^*$ is the simplicial completion of the original Outer space. The resulting picture fits together many familiar objects from geometric group theory and surface topology, including Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichmüller space. Many of the relationships between the objects that we see in this picture already exist in the literature, but we add some new ones, and generalize several existing relationships to include a number $n>0$ of punctures/leaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Handlebodies, Outer space, and tropical geometry Ramadas, Rohini Silversmith, Rob Vogtmann, Karen Winarski, Rebecca R. Geometric Topology Algebraic Geometry Group Theory 14T20, 20F65, 57K20, 32G15 The moduli space of graphs $M_{g,n}^{\mathrm{trop}}$ is a polyhedral object that mimics the behavior of the moduli spaces $M_{g,n}$, $\overline{M}_{g,n}$ of (stable) Riemann surfaces; this relationship has been made precise in several different ways, which collectively identify $M_{g,n}^{\mathrm{trop}}$ as the "tropicalization" of $M_{g,n}$. We describe how this relationship lifts to some objects that live over $M_{g,n}$ (like Teichmüller space) and that live over $M_{g,n}^{\mathrm{trop}}$ (like the Culler-Vogtmann space $CV_{g,n}^*$). We introduce the notion of a stable complex handlebody, and show that $CV_{g,n}^*$ can be viewed as the tropicalization of a certain complex manifold $hT(V_{g,n})$ that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification $\overline{hT}(V_{g,n})\supset hT(V_{g,n})$, which we prove is a simply connected complex manifold with simple normal crossings boundary. When $n=0$, $hT(V_{g,n})$ coincides with the moduli space of Schottky groups, $\overline{hT}(V_{g,n})$ coincides with Gerritzen-Herrlich's extended Schottky space, and $CV_{g,0}^*$ is the simplicial completion of the original Outer space. The resulting picture fits together many familiar objects from geometric group theory and surface topology, including Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichmüller space. Many of the relationships between the objects that we see in this picture already exist in the literature, but we add some new ones, and generalize several existing relationships to include a number $n>0$ of punctures/leaves. |
| title | Handlebodies, Outer space, and tropical geometry |
| topic | Geometric Topology Algebraic Geometry Group Theory 14T20, 20F65, 57K20, 32G15 |
| url | https://arxiv.org/abs/2507.02440 |