Geodesic
Girth Alternative for subgroups of $PL_o(I)$
Glasgow mathematical journal, Tome 67 (2025) no. 1, pp. 1-10

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DOI

We prove the Girth Alternative for finitely generated subgroups of $PL_o(I)$. We also prove that a finitely generated subgroup of Homeo$_{+}(I)$ which is sufficiently rich with hyperbolic-like elements has infinite girth. 
Akhmedov, Azer. Girth Alternative for subgroups of $PL_o(I)$. Glasgow mathematical journal, Tome 67 (2025) no. 1, pp. 1-10. doi: 10.1017/S0017089524000181
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     title = {Girth {Alternative} for subgroups of $PL_o(I)$},
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