A realisation result for moduli spaces of group actions on the line
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917764328849408 |
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| author | Brum, Joaquín Bon, Nicolás Matte Rivas, Cristóbal Triestino, Michele |
| author_facet | Brum, Joaquín Bon, Nicolás Matte Rivas, Cristóbal Triestino, Michele |
| contents | Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $(Y, Φ)$ naturally associated with $G$ and uniquely defined up to flow equivalence, that we call the \emph{Deroin space} of $G$. We show a realisation result: every expansive flow $(Y, Φ)$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_03846 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A realisation result for moduli spaces of group actions on the line Brum, Joaquín Bon, Nicolás Matte Rivas, Cristóbal Triestino, Michele Group Theory Dynamical Systems Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $(Y, Φ)$ naturally associated with $G$ and uniquely defined up to flow equivalence, that we call the \emph{Deroin space} of $G$. We show a realisation result: every expansive flow $(Y, Φ)$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions. |
| title | A realisation result for moduli spaces of group actions on the line |
| topic | Group Theory Dynamical Systems |
| url | https://arxiv.org/abs/2306.03846 |