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Timeline for Is Higman's group a free product?

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Jan 15, 2021 at 16:28 comment added Guest Thanks! This is very informative.
Jan 13, 2021 at 9:46 comment added YCor I think the argument shows that whenever $H$ acts isometrically on a (nonempty) Gromov-hyperbolic space, either the action is horocyclic (there's a unique fixed point at infinity and no loxodromic), or each generator, actually each $H_i$, acts with bounded orbits. In particular every proper such action is horocyclic (while an infinite relatively hyperbolic group always admits a proper non-horocyclic action).
Jan 13, 2021 at 9:32 history edited YCor CC BY-SA 4.0
emphasized main statement to help reader (feel free to revert)
Jan 13, 2021 at 8:38 history answered AGenevois CC BY-SA 4.0