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Quasi-isometric embedding from the generalised Thompson's group to
Brown has defined the generalised Thompson's group , , where is
an integer at least and Thompson's groups and in the
80's. Burillo, Cleary and Stein have found that there is a quasi-isometric
embedding from to where and are positive integers at least
2. We show that there is a quasi-isometric embedding from to for
any and no embeddings from to for
On planar Cayley graphs and Kleinian groups
Let be a finitely generated group acting faithfully and properly
discontinuously by homeomorphisms on a planar surface . We prove that admits such an action that is in addition
co-compact, provided we can replace by another surface .
We also prove that if a group has a finitely generated Cayley
(multi-)graph covariantly embeddable in , then can be
chosen so as to have no infinite path on the boundary of a face.
The proofs of these facts are intertwined, and the classes of groups they
define coincide. In the orientation-preserving case they are exactly the
(isomorphism types of) finitely generated Kleinian function groups. We
construct a finitely generated planar Cayley graph whose group is not in this
class.
In passing, we observe that the Freudenthal compactification of every planar
surface is homeomorphic to the sphere
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