Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2411.04921 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912109443416064 |
|---|---|
| author | Monti, Andrea Egidio |
| author_facet | Monti, Andrea Egidio |
| contents | Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation surfaces. Thurston established an equivalence between complex projective structures and hyperbolic surfaces grafted along a measured lamination, leading to a homeomorphism $\mathcal{PT}(S) \cong \mathcal{T}(S) \times \mathcal{ML}(S)$. Our bordification is geometric in the sense that convergence to points on the boundary corresponds to the geometric convergence of grafted surfaces to half-translation surfaces (up to rescaling). This result relies on recent work by Calderon and Farre on the orthogeodesic foliation construction. Finally, we introduce a change of perspective, viewing grafted surfaces as a deformation (which we term "inflation") of half-translation surfaces, consisting of inserting negatively curved regions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric boundary for the moduli space of grafted surfaces Monti, Andrea Egidio Geometric Topology Let $S$ be a closed orientable surface of genus at least two. We introduce a bordification of the moduli space $\mathcal{PT}(S)$ of complex projective structures, with a boundary consisting of projective classes of half-translation surfaces. Thurston established an equivalence between complex projective structures and hyperbolic surfaces grafted along a measured lamination, leading to a homeomorphism $\mathcal{PT}(S) \cong \mathcal{T}(S) \times \mathcal{ML}(S)$. Our bordification is geometric in the sense that convergence to points on the boundary corresponds to the geometric convergence of grafted surfaces to half-translation surfaces (up to rescaling). This result relies on recent work by Calderon and Farre on the orthogeodesic foliation construction. Finally, we introduce a change of perspective, viewing grafted surfaces as a deformation (which we term "inflation") of half-translation surfaces, consisting of inserting negatively curved regions. |
| title | A geometric boundary for the moduli space of grafted surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2411.04921 |