Abstract We prove the Girth Alternative for finitely generated subgroups of PLₒ (I). We also prove that a finitely generated subgroup of Homeo + (I) which is sufficiently rich with hyperbolic-like elements has infinite girth.

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Azer Akhmedov (Thu,) studied this question.

synapsesocial.com/papers/68e6aeb0b6db643587630939 https://doi.org/https://doi.org/10.1017/s0017089524000181

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Also Consider

Synapse has enriched 4 closely related papers on similar clinical questions. Consider them for comparative context:

1Elementary amenable subgroups of R. Thompson’s group F2012 · 31 citations
2ELEMENTARY AMENABLE SUBGROUPS OF R. THOMPSON'S GROUP F2005 · 17 citations
3A geometric classification of some solvable groups of homeomorphisms2008 · 26 citations
4On the girth of finitely generated groups2003 · 12 citations