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Nov 15, 2023 at 17:46 comment added Ashot Minasyan @YanlongHao If you do not require properness of the action then, I think, even for large $n$ it is still possible to construct counter-examples among groups acting on trees.
Nov 9, 2023 at 23:47 comment added YCor "If $\Gamma$ is not torsion-free one has to assume at least $g\notin E(\gamma)$": this is of course not the case if $g,\gamma$ freely generate a semigroup. But this assumption was added after this answer was posted.
Nov 9, 2023 at 13:03 comment added Yanlong Hao Thanks for your answer and reference. I want to consider the solution for $n$ big enough. But you example is very interesting.
Nov 9, 2023 at 12:41 vote accept Yanlong Hao
Nov 9, 2023 at 11:31 history answered Ashot Minasyan CC BY-SA 4.0