Timeline for Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective?
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| Aug 3, 2023 at 17:16 | comment | added | YCor | For your second question, taking $H=\mathrm{Isom}(G)$, you're asking whether $W$ is trivial... no, just take a trivalent tree with 2 hair at each vertex. | |
| Aug 3, 2023 at 17:15 | comment | added | YCor | In (d), (e), there's a loxodromic but a single global fixed point. So there are at least 3 boundary points. So the loxodromic moves one of them. Hence the action on the boundary is nontrivial. | |
| Aug 3, 2023 at 8:43 | comment | added | John Depp | Thank you for yor answer! Could you elaborate a bit more why " (d), (e) are excluded " ? I have another question related to this statement. Suppose a subgroup $H$ of $Isom(X)$ is acting non-elementarily on $X$ ( I guess this is same as option (d) and (e) ). Is it true that the intersection of $W$ and $H$ is trivial ? | |
| Aug 2, 2023 at 14:16 | history | answered | YCor | CC BY-SA 4.0 |