Distorted and undistorted subgroups of the Lodha-Moore group

We show that the Baumslag-Solitar group $BS(1,2)$ is undistorted in the Lodha-Moore group $G_0$ using an explicit lower bound for the word length of $G_0$. We also show that Thompson’s group $F$ is distorted in $G_0$.

💡 Research Summary

This paper investigates the geometric properties of the Lodha–Moore group G₀, focusing on the distortion behavior of two well‑studied subgroups: the Baumslag–Solitar group BS(1, 2) and Thompson’s group F. The authors begin by recalling the definition of G₀ as a finitely generated subgroup of the piecewise‑projective homeomorphism group PPSL₂(ℝ). The three standard generators a(t)=t+1, b(t), and c(t) are described explicitly; each acts as a simple translation or a piecewise linear‑fractional map on specific intervals. G₀ is known to be finitely presented, torsion‑free, non‑amenable, and of type F_∞, sharing many combinatorial features with Thompson’s group F while differing dramatically in amenability.

To quantify the complexity of an element f∈G₀, the authors introduce two integer‑valued invariants. The first, D(f), records the maximal denominator appearing among the rational break points of f (points where the piecewise definition changes). The second, M(f), records the maximal absolute entry among the integer matrices that describe f on each linear‑fractional component. The combined invariant C(f)=max{D(f), M(f)} serves as a measure of “size” for f. A series of lemmas (3.1–3.5) shows that multiplying f on the right by any generator s∈S_{G₀}={a^{±1},b^{±1},c^{±1}} inflates C at most by a constant factor: C(fs)≤6 C(f). Iterating this bound yields a global lower bound for word length: \