Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups
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| Format: | Preprint |
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2024
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| _version_ | 1866914177948319744 |
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| author | Kineider, Clarence Troubat, Roméo |
| author_facet | Kineider, Clarence Troubat, Roméo |
| contents | We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and, using methods of Dey--Greenberg--Riestenberg, we show that for certain parabolic subgroups $P_Θ$, any $P_Θ$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit computation of some Plücker coordinates to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $Θ$-positive semigroup $U_Θ^{>0}$ when $p\neq q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_08679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups Kineider, Clarence Troubat, Roméo Differential Geometry Group Theory Geometric Topology Representation Theory We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and, using methods of Dey--Greenberg--Riestenberg, we show that for certain parabolic subgroups $P_Θ$, any $P_Θ$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit computation of some Plücker coordinates to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $Θ$-positive semigroup $U_Θ^{>0}$ when $p\neq q$. |
| title | Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups |
| topic | Differential Geometry Group Theory Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2411.08679 |