Standard-action conjecture for over rings of integers
Let be a number field with ring of integers , and set . The group is a lattice in
where and are the numbers of real and complex places of , respectively. A hyperbolic -space is standard if it is equivalent to or with one of the actions obtained by projection to a real or complex factor. A quasi-action is an action by quasi-isometries satisfying the group-action laws up to uniformly bounded error.
Standard-action conjecture. Every quasi-action by on a Gromov hyperbolic metric space either has an invariant horoball or is standard.
This conjecture predicts that, apart from actions preserving a horoball, all quasi-actions of on Gromov hyperbolic spaces arise from the natural actions on the hyperbolic planes or hyperbolic three-spaces associated with the archimedean factors. The supplied source does not state whether the conjecture is resolved.
References
Primary source
Jason Fox Manning, “Actions of certain arithmetic groups on Gromov hyperbolic spaces”, arXiv:math/0702749 (2008).
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