Subspace stabilisers in hyperbolic lattices
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2021
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918190893760512 |
|---|---|
| author | Belolipetsky, Mikhail Bogachev, Nikolay Kolpakov, Alexander Slavich, Leone |
| author_facet | Belolipetsky, Mikhail Bogachev, Nikolay Kolpakov, Alexander Slavich, Leone |
| contents | This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \frac{n-1}{2}$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices.
We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, and exhibit examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterization of totally geodesically immersed suborbifolds of arithmetic hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This allows us to construct examples with the property that the adjoint trace field of the geodesic suborbifold properly contains the adjoint trace field of the orbifold. The case of special interest is that of exceptional trialitarian $7$-dimensional orbifolds. We show that every such orbifold contains a totally geodesic arithmetic hyperbolic $3$-orbifold of exceptional type.
Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds, proving that all suborbifolds in a (quasi-)arithmetic orbifold are (quasi-)arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_06897 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Subspace stabilisers in hyperbolic lattices Belolipetsky, Mikhail Bogachev, Nikolay Kolpakov, Alexander Slavich, Leone Geometric Topology Differential Geometry Group Theory Metric Geometry Number Theory This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \frac{n-1}{2}$. We call such totally geodesic suborbifolds finite centraliser subspaces (or fc-subspaces) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold $M$ is arithmetic if and only if it has infinitely many fc-subspaces, and exhibit examples of non-arithmetic orbifolds that contain non-fc subspaces of codimension one. We provide an algebraic characterization of totally geodesically immersed suborbifolds of arithmetic hyperbolic orbifolds by analysing Vinberg's commensurability invariants. This allows us to construct examples with the property that the adjoint trace field of the geodesic suborbifold properly contains the adjoint trace field of the orbifold. The case of special interest is that of exceptional trialitarian $7$-dimensional orbifolds. We show that every such orbifold contains a totally geodesic arithmetic hyperbolic $3$-orbifold of exceptional type. Finally, we study arithmetic properties of orbifolds that descend to their totally geodesic suborbifolds, proving that all suborbifolds in a (quasi-)arithmetic orbifold are (quasi-)arithmetic. |
| title | Subspace stabilisers in hyperbolic lattices |
| topic | Geometric Topology Differential Geometry Group Theory Metric Geometry Number Theory |
| url | https://arxiv.org/abs/2105.06897 |