Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918107744829440 |
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| author | Jaeck, Victor |
| author_facet | Jaeck, Victor |
| contents | We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $Ξ\times Y \rightarrow Ξ$ extends to a $Γ$-equivariant continuous surjection $(Ξ\times Y)^{\mathrm{RSp}} \rightarrow Ξ^{\mathrm{RSp}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{RSp}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y=\mathbb{P}^1$, the image is a real subtree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22654 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties Jaeck, Victor Group Theory Algebraic Geometry Differential Geometry Geometric Topology 22E40, 14P10, 12J15, 12J25, 14G22 We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $Ξ\times Y \rightarrow Ξ$ extends to a $Γ$-equivariant continuous surjection $(Ξ\times Y)^{\mathrm{RSp}} \rightarrow Ξ^{\mathrm{RSp}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{RSp}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y=\mathbb{P}^1$, the image is a real subtree. |
| title | Real Spectrum Compactifications of Universal Geometric Spaces over Character Varieties |
| topic | Group Theory Algebraic Geometry Differential Geometry Geometric Topology 22E40, 14P10, 12J15, 12J25, 14G22 |
| url | https://arxiv.org/abs/2507.22654 |