On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917572351361024 |
|---|---|
| author | Aiello, Valeriano Nagnibeda, Tatiana |
| author_facet | Aiello, Valeriano Nagnibeda, Tatiana |
| contents | In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we continue with the study of the $3$-colorable subgroup and prove that the quasi-regular representation of $F$ associated with the $3$-colorable subgroup is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a certain injective endomorphism of $F$ is contained in three (explicit) maximal subgroups of $F$ of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of $F$, namely the parabolic subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented subgroup $\vec{F}$, and the explicit examples found by Golan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_07885 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$ Aiello, Valeriano Nagnibeda, Tatiana Group Theory Operator Algebras In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we continue with the study of the $3$-colorable subgroup and prove that the quasi-regular representation of $F$ associated with the $3$-colorable subgroup is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a certain injective endomorphism of $F$ is contained in three (explicit) maximal subgroups of $F$ of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of $F$, namely the parabolic subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented subgroup $\vec{F}$, and the explicit examples found by Golan. |
| title | On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$ |
| topic | Group Theory Operator Algebras |
| url | https://arxiv.org/abs/2103.07885 |