Timeline for Ping pong with parabolic isometries on Gromov hyperbolic spaces
Current License: CC BY-SA 4.0
8 events
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| May 27, 2021 at 11:40 | comment | added | NWMT | I'm reading a bit more closely and the classification of isometries requires $X$ to be proper, but if you already know your isometries are parabolic, then I think a metric argument will work. | |
| May 27, 2021 at 11:26 | comment | added | NWMT | I'm not sure such a result exists for arbitrary (i.e. non proper) Gromov hyperbolic spaces. There exists an argument that uses only metric properties of $\partial X$, but the source I have, Ghys, Étienne. "Sur les groupes hyperboliques d'après Mikhael Gromov." Progr. Math. 83 (1990), assumes $X$ is proper. | |
| May 26, 2021 at 11:35 | comment | added | NWMT | Are you requiring that $X$ is a proper metric space or are you allowing something more exotic, like an $\mathbb R$-tree? If $X$ is a proper metric space (i.e. locally compact) then $\partial X$ is compact and metrizable and it's just a standard ping pong argument using disjoint neighbourhoods of the limit points, the fact that their closures are compact, and the definition of a parabolic element here: en.wikipedia.org/wiki/Convergence_group | |
| May 5, 2021 at 19:33 | comment | added | Richard Weidmann | Not a real answer. Lemme 2.3 of the book of Coornaert, Delzant and Papadopoulos gives a criterion when the product of two non-hyperbolic isometries is hyperbolic. In you case this should prove that for sufficiently large $M$ the product $f^{\pm M}g^{\pm M}$ is hyperbolic. With a little bit of work you should be able to make this argument work for any freely reduced product in $f^M$ and $g^M$. | |
| May 4, 2021 at 10:51 | comment | added | YCor | It seems to me that (3) follows from (1), in an arbitrary proper Gromov-hyperbolic space (in a proper metric space $X$ with isometry $f$, $(d(o,f^n(o)))_n$ is either bounded or tends to infinity. | |
| May 4, 2021 at 10:49 | history | edited | YCor |
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| May 4, 2021 at 10:45 | review | First posts | |||
| May 4, 2021 at 11:00 | |||||
| May 4, 2021 at 10:44 | history | asked | user203667 | CC BY-SA 4.0 |