The poset of hyperbolic structures on F_n has two maximal quasi-parabolic elements and uncountably many lamplike subposets described via lamplighter groups, all collapsing under semidirect product with Z/2Z.
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Hyperbolic actions of Thompson's group $F$ and generalizations
The poset of hyperbolic structures on F_n has two maximal quasi-parabolic elements and uncountably many lamplike subposets described via lamplighter groups, all collapsing under semidirect product with Z/2Z.