Real spectrum and oriented Gromov equivariant compactifications of character varieties
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909796265885696 |
|---|---|
| author | Jaeck, Victor |
| author_facet | Jaeck, Victor |
| contents | The character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{PSL}_2(\mathbb{R})$ has many compactifications. We construct a continuous surjection from the real spectrum compactification $Ξ^{\mathrm{RSp}}$ to the oriented Gromov equivariant compactification. Our construction is based on a geometric interpretation of the elements of $\partial Ξ^{\mathrm{RSp}}$ as $Γ$-actions by isometries on $\mathbb{R}$-trees. We endow these $\mathbb{R}$-trees with an orientation induced by the standard orientation on the circle, which we characterize by a semialgebraic equation. Moreover, we describe the $Γ$-actions by orientation preserving isometries on oriented $\mathbb{R}$-trees, which arise in both compactifications, as limits of $Γ$-actions on the oriented hyperbolic plane, via asymptotic cones endowed with an ultralimit orientation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Real spectrum and oriented Gromov equivariant compactifications of character varieties Jaeck, Victor Geometric Topology Algebraic Geometry Group Theory The character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{PSL}_2(\mathbb{R})$ has many compactifications. We construct a continuous surjection from the real spectrum compactification $Ξ^{\mathrm{RSp}}$ to the oriented Gromov equivariant compactification. Our construction is based on a geometric interpretation of the elements of $\partial Ξ^{\mathrm{RSp}}$ as $Γ$-actions by isometries on $\mathbb{R}$-trees. We endow these $\mathbb{R}$-trees with an orientation induced by the standard orientation on the circle, which we characterize by a semialgebraic equation. Moreover, we describe the $Γ$-actions by orientation preserving isometries on oriented $\mathbb{R}$-trees, which arise in both compactifications, as limits of $Γ$-actions on the oriented hyperbolic plane, via asymptotic cones endowed with an ultralimit orientation. |
| title | Real spectrum and oriented Gromov equivariant compactifications of character varieties |
| topic | Geometric Topology Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2402.06444 |