$Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909459192741888 |
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| author | Osin, D. |
| author_facet | Osin, D. |
| contents | Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic, $Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the other hand, there are non-empty closed subsets of $\mathcal G_n$ that admit no $Out(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the group $Aut(F_n)$ plays a crucial role in the proof of both results. We also discuss model theoretic implications of the existence of $Out(F_n)$-invariant, ergodic probability measures on $\mathcal G_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03659 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups Osin, D. Group Theory Dynamical Systems Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic, $Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the other hand, there are non-empty closed subsets of $\mathcal G_n$ that admit no $Out(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the group $Aut(F_n)$ plays a crucial role in the proof of both results. We also discuss model theoretic implications of the existence of $Out(F_n)$-invariant, ergodic probability measures on $\mathcal G_n$. |
| title | $Out(F_n)$-invariant probability measures on the space of $n$-generated marked groups |
| topic | Group Theory Dynamical Systems |
| url | https://arxiv.org/abs/2207.03659 |